Difference between revisions of "Method of Undetermined Coefficients"

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In mathematics, the '''method of undetermined coefficients''' is an approach to finding a particular solution to certain nonhomogeneous [[ordinary differential equation]]s and [[recurrence relation]]s. It is closely related to the [[annihilator method]], but instead of using a particular kind of [[differential operator]] (the annihilator) in order to find the best possible form of the particular solution, a "guess" is made as to the appropriate form, which is then tested by differentiating the resulting equation. For complex equations, the annihilator method or [[variation of parameters]] is less time-consuming to perform.
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In mathematics, the '''method of undetermined coefficients''' is an approach to finding a particular solution to certain nonhomogeneous ordinary differential equations and recurrence relations. It is closely related to the annihilator method, but instead of using a particular kind of differential operator (the annihilator) in order to find the best possible form of the particular solution, a "guess" is made as to the appropriate form, which is then tested by differentiating the resulting equation. For complex equations, the annihilator method or variation of parameters is less time-consuming to perform.
  
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Undetermined coefficients is not as general a method as [[variation of parameters]], since it only works for differential equations that follow certain forms.<ref name="Grimaldi">Ralph P. Grimaldi (2000). "Nonhomogeneous Recurrence Relations". Section 3.3.3 of ''Handbook of Discrete and Combinatorial Mathematics''. Kenneth H. Rosen, ed. CRC Press. {{isbn|0-8493-0149-1}}.</ref>
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Undetermined coefficients is not as general a method as variation of parameters, since it only works for differential equations that follow certain forms.
  
 
==Description of the method==
 
==Description of the method==
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:where <math>y^{(i)}</math> denotes the i-th derivative of <math>y</math>, and <math>c_i</math> denotes a function of <math>x</math>.
 
:where <math>y^{(i)}</math> denotes the i-th derivative of <math>y</math>, and <math>c_i</math> denotes a function of <math>x</math>.
  
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The method of undetermined coefficients provides a straightforward method of obtaining the solution to this ODE when two criteria are met:<ref>{{Cite book|last=Zill, Dennis G., Warren S. Wright|title=Advanced Engineering Mathematics|publisher=Jones and Bartlett|year=2014|isbn=978-1-4496-7977-4|pages=125}}</ref>
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The method of undetermined coefficients provides a straightforward method of obtaining the solution to this ODE when two criteria are met:
 
# <math>c_i</math> are constants.
 
# <math>c_i</math> are constants.
 
# ''g''(''x'') is a constant, a polynomial function, exponential function <math>e^{\alpha x}</math>, sine or cosine functions <math>\sin{\beta x}</math> or <math>\cos{\beta x}</math>, or finite sums and products of these functions (<math>{\alpha}</math>, <math>{\beta}</math> constants).
 
# ''g''(''x'') is a constant, a polynomial function, exponential function <math>e^{\alpha x}</math>, sine or cosine functions <math>\sin{\beta x}</math> or <math>\cos{\beta x}</math>, or finite sums and products of these functions (<math>{\alpha}</math>, <math>{\beta}</math> constants).
  
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The method consists of finding the general [[Homogeneous differential equation|homogeneous]] solution <math>y_c</math> for the complementary linear [[homogeneous differential equation]]
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The method consists of finding the general homogeneous solution <math>y_c</math> for the complementary linear homogeneous differential equation
  
 
:<math> \sum_{i=0}^n c_i y^{(i)} + y^{(n+1)} = 0,</math>
 
:<math> \sum_{i=0}^n c_i y^{(i)} + y^{(n+1)} = 0,</math>
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and a particular integral <math>y_p</math> of the linear non-homogeneous ordinary differential equation based on <math>g(x)</math>. Then the general solution <math>y</math> to the linear non-homogeneous ordinary differential equation would be
 
and a particular integral <math>y_p</math> of the linear non-homogeneous ordinary differential equation based on <math>g(x)</math>. Then the general solution <math>y</math> to the linear non-homogeneous ordinary differential equation would be
  
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:<math>y = y_c + y_p.</math><ref name="Zill2008">{{cite book|author=Dennis G. Zill|title=A First Course in Differential Equations|url=https://books.google.com/books?id=BnArjLNjXuYC&q=%22undetermined+coefficients%22|date=14 May 2008|publisher=Cengage Learning|isbn=978-0-495-10824-5}}</ref>
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:<math>y = y_c + y_p.</math>
  
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If <math>g(x)</math> consists of the sum of two functions <math>h(x) + w(x)</math> and we say that <math>y_{p_1}</math> is the solution based on <math>h(x)</math> and <math> y_{p_2}</math> the solution based on <math>w(x)</math>. Then, using a [[superposition principle]], we can say that the particular integral <math>y_p</math> is<ref name="Zill2008" />
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If <math>g(x)</math> consists of the sum of two functions <math>h(x) + w(x)</math> and we say that <math>y_{p_1}</math> is the solution based on <math>h(x)</math> and <math> y_{p_2}</math> the solution based on <math>w(x)</math>. Then, using a superposition principle, we can say that the particular integral <math>y_p</math> is
  
 
:<math>y_p = y_{p_1} + y_{p_2}.</math>
 
:<math>y_p = y_{p_1} + y_{p_2}.</math>
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|}
 
|}
  
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If a term in the above particular integral for ''y'' appears in the homogeneous solution, it is necessary to multiply by a sufficiently large power of ''x'' in order to make the solution independent. If the function of ''x'' is a sum of terms in the above table, the particular integral can be guessed using a sum of the corresponding terms for ''y''.<ref name="Grimaldi"/>
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If a term in the above particular integral for ''y'' appears in the homogeneous solution, it is necessary to multiply by a sufficiently large power of ''x'' in order to make the solution independent. If the function of ''x'' is a sum of terms in the above table, the particular integral can be guessed using a sum of the corresponding terms for ''y''.
  
 
==Examples==
 
==Examples==
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<p>We will now look at this method for higher order linear nonhomogenous differential equations. Consider the following <math>n^{\mathrm{th}}</math> order linear nonhomogenous differential equation with coefficients <math>a_0, a_1, ..., a_n \in \mathbb{R}</math>:</p>
 
<p>We will now look at this method for higher order linear nonhomogenous differential equations. Consider the following <math>n^{\mathrm{th}}</math> order linear nonhomogenous differential equation with coefficients <math>a_0, a_1, ..., a_n \in \mathbb{R}</math>:</p>
  
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<math>\begin{align} \quad a_0 \frac{d^ny}{dt^n} + a_1 \frac{d^{n-1}y}{dt^{n-1}} + ... + a_{n-1}\frac{dy}{dt} + a_n y = g(t) \end{align}</math>
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<math>\begin{align} \quad a_0 \frac{d^ny}{dt^n} + a_1 \frac{d^{n-1}y}{dt^{n-1}} + ... + a_{n-1}\frac{dy}{dt} + a_n y = g(t) \end{align} </math>
 
<p>Suppose that <math>g(t)</math> is of a form containing a polynomial, exponential function, or a sine/cosine function (like with when we were dealing with the method of undetermined coefficients for second order linear nonhomogenous differential equations). We can then find a particular solution <math>Y(t)</math> if we extend the assumed forms of combinations of polynomials, exponential functions, or sine/cosine functions, and multiplying by powers of <math>t</math> to ensure our particular solution does not contain part of a solution to the corresponding higher order linear homogenous differential equation. We can then solve for these constants by plugging our assumed form <math>Y(t)</math> into our differential equation above.</p>
 
<p>Suppose that <math>g(t)</math> is of a form containing a polynomial, exponential function, or a sine/cosine function (like with when we were dealing with the method of undetermined coefficients for second order linear nonhomogenous differential equations). We can then find a particular solution <math>Y(t)</math> if we extend the assumed forms of combinations of polynomials, exponential functions, or sine/cosine functions, and multiplying by powers of <math>t</math> to ensure our particular solution does not contain part of a solution to the corresponding higher order linear homogenous differential equation. We can then solve for these constants by plugging our assumed form <math>Y(t)</math> into our differential equation above.</p>
 
<p>Providing a list of possible forms is trivial, so we will instead look at some examples of apply this method.</p>
 
<p>Providing a list of possible forms is trivial, so we will instead look at some examples of apply this method.</p>

Latest revision as of 14:54, 26 October 2021

In mathematics, the method of undetermined coefficients is an approach to finding a particular solution to certain nonhomogeneous ordinary differential equations and recurrence relations. It is closely related to the annihilator method, but instead of using a particular kind of differential operator (the annihilator) in order to find the best possible form of the particular solution, a "guess" is made as to the appropriate form, which is then tested by differentiating the resulting equation. For complex equations, the annihilator method or variation of parameters is less time-consuming to perform.

Undetermined coefficients is not as general a method as variation of parameters, since it only works for differential equations that follow certain forms.

Description of the method

Consider a linear non-homogeneous ordinary differential equation of the form

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{i=0}^n c_i y^{(i)} + y^{(n+1)} = g(x)}
where denotes the i-th derivative of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y} , and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_i} denotes a function of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} .

The method of undetermined coefficients provides a straightforward method of obtaining the solution to this ODE when two criteria are met:

  1. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_i} are constants.
  2. g(x) is a constant, a polynomial function, exponential function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e^{\alpha x}} , sine or cosine functions Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin{\beta x}} or Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cos{\beta x}} , or finite sums and products of these functions (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\alpha}} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\beta}} constants).

The method consists of finding the general homogeneous solution Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_c} for the complementary linear homogeneous differential equation

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{i=0}^n c_i y^{(i)} + y^{(n+1)} = 0,}

and a particular integral Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_p} of the linear non-homogeneous ordinary differential equation based on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g(x)} . Then the general solution Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y} to the linear non-homogeneous ordinary differential equation would be

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y = y_c + y_p.}

If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g(x)} consists of the sum of two functions Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h(x) + w(x)} and we say that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_{p_1}} is the solution based on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h(x)} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_{p_2}} the solution based on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle w(x)} . Then, using a superposition principle, we can say that the particular integral Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_p} is

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_p = y_{p_1} + y_{p_2}.}

Typical forms of the particular integral

In order to find the particular integral, we need to 'guess' its form, with some coefficients left as variables to be solved for. This takes the form of the first derivative of the complementary function. Below is a table of some typical functions and the solution to guess for them.

Function of x Form for y
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k e^{a x}\!} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C e^{a x}\!}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k x^n,\; n = 0, 1, 2,\ldots\!}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{i=0}^n K_i x^i \!}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k \cos(a x) \text{ or } k \sin(a x) \!}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K \cos(a x) + M \sin(a x) \!}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k e^{a x} \cos(b x) \text{ or } ke^{a x} \sin(b x) \!}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e^{a x} (K \cos(b x) + M \sin(b x)) \!}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left(\sum_{i=0}^n k_i x^i\right) \cos(b x) \text{ or }\ \left(\sum_{i=0}^n k_i x^i\right) \sin(b x) \!}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left(\sum_{i=0}^n Q_i x^i\right) \cos(b x) + \left(\sum_{i=0}^n R_i x^i\right) \sin(b x)}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left(\sum_{i=0}^n k_i x^i\right) e^{a x} \cos(b x) \text{ or } \left(\sum_{i=0}^n k_i x^i\right) e^{a x} \sin(b x)\!}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e^{a x} \left(\left(\sum_{i=0}^n Q_i x^i\right) \cos(b x) + \left(\sum_{i=0}^n R_i x^i\right) \sin(b x)\right)}

If a term in the above particular integral for y appears in the homogeneous solution, it is necessary to multiply by a sufficiently large power of x in order to make the solution independent. If the function of x is a sum of terms in the above table, the particular integral can be guessed using a sum of the corresponding terms for y.

Examples

Example 1

Find a particular integral of the equation

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y'' + y = t \cos t. }

The right side t cos t has the form

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_n e^{\alpha t} \cos{\beta t} }

with n = 2, α = 0, and β = 1.

Since α + iβ = i is a simple root of the characteristic equation

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda^2 + 1 = 0 }

we should try a particular integral of the form

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} y_p &= t \left [F_1 (t) e^{\alpha t} \cos{\beta t} + G_1 (t) e^{\alpha t} \sin{\beta t} \right ] \\ &= t \left [F_1 (t) \cos t + G_1 (t) \sin t \right ] \\ &= t \left [ \left (A_0 t + A_1 \right ) \cos t + \left (B_0 t + B_1 \right ) \sin t \right ] \\ &= \left (A_0 t^2 + A_1 t \right ) \cos t + \left (B_0 t^2 + B_1 t \right) \sin t. \end{align}}

Substituting yp into the differential equation, we have the identity

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} t \cos t &= y_p'' + y_p \\ &= \left [ \left(A_0 t^2 + A_1 t \right ) \cos t + \left (B_0 t^2 + B_1 t \right ) \sin t \right ]'' + \left[\left(A_0 t^2 + A_1 t \right ) \cos t + \left(B_0 t^2 + B_1 t \right ) \sin t \right ] \\ &= \left [2A_0 \cos t + 2 \left (2A_0 t + A_1 \right )(-\sin t) + \left (A_0 t^2 + A_1 t \right )(-\cos t) + 2B_0 \sin t + 2 \left (2B_0 t + B_1 \right ) \cos t + \left (B_0 t^2 + B_1 t \right )(- \sin t) \right ] \\ &\qquad +\left[\left(A_0 t^2 + A_1 t \right ) \cos t + \left(B_0 t^2 + B_1 t \right ) \sin t \right ] \\ &= [4B_0 t + (2A_0 + 2B_1)] \cos t + [-4A_0 t + (-2A_1 + 2B_0)] \sin t. \end{align}}

Comparing both sides, we have

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{cases} 1 = 4B_0\\ 0 = 2A_0 + 2B_1 \\ 0 = -4A_0 \\ 0 = -2A_1 + 2B_0 \end{cases}}

which has the solution

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A_0 = 0, \quad A_1 = B_0 = \frac{1}{4}, \quad B_1 = 0.}

We then have a particular integral

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_p = \frac {1} {4} t \cos t + \frac {1}{4} t^2 \sin t. }

Example 2

Consider the following linear nonhomogeneous differential equation:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{dy}{dx} = y + e^x.}

This is like the first example above, except that the nonhomogeneous part (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e^x} ) is not linearly independent to the general solution of the homogeneous part (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_1 e^x} ); as a result, we have to multiply our guess by a sufficiently large power of x to make it linearly independent.

Here our guess becomes:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_p = A x e^x.}

By substituting this function and its derivative into the differential equation, one can solve for A:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{d}{dx} \left( A x e^x \right) = A x e^x + e^x}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A x e^x + A e^x = A x e^x + e^x}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A = 1.}

So, the general solution to this differential equation is:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y = c_1 e^x + xe^x.}

Example 3

Find the general solution of the equation:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{dy}{dt} = t^2 - y}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t^2} is a polynomial of degree 2, so we look for a solution using the same form,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_p = A t^2 + B t + C,}

Plugging this particular function into the original equation yields,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2 A t + B = t^2 - (A t^2 + B t + C),}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2 A t + B =(1-A)t^2 -Bt -C, }
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (A-1)t^2 + (2A+B)t + (B+C) = 0.}

which gives:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A-1 = 0, \quad 2A+B =0, \quad B+C=0.}

Solving for constants we get:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_p = t^2 - 2 t + 2}

To solve for the general solution,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y= y_p + y_c}

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_c} is the homogeneous solution Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_c = c_1 e^{-t}} , therefore, the general solution is:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y= t^2 - 2 t + 2 + c_1 e^{-t}}

Higher Order

If we have a second order linear nonhomogenous differential equation whose coefficients were constant, that is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \frac{d^2y}{dt^2} + b \frac{dy}{dt} + cy = g(t)} , then to solve this differential equation, all we need to do is solve the corresponding second order linear homogenous differential equation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \frac{d^2y}{dt^2} + b \frac{dy}{dt} + cy = 0} , and then find a partial solution by assuming the form of the particular solution. More precisely, if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g(t)} was a polynomial, exponential, or sine/cosine function (or a combination of these), then we could assume a form for the particular solutions (see the linked page above for more details) and solve for the coefficients of this form to obtain a particular solution.

We will now look at this method for higher order linear nonhomogenous differential equations. Consider the following Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n^{\mathrm{th}}} order linear nonhomogenous differential equation with coefficients Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_0, a_1, ..., a_n \in \mathbb{R}} :

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \quad a_0 \frac{d^ny}{dt^n} + a_1 \frac{d^{n-1}y}{dt^{n-1}} + ... + a_{n-1}\frac{dy}{dt} + a_n y = g(t) \end{align} }

Suppose that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g(t)} is of a form containing a polynomial, exponential function, or a sine/cosine function (like with when we were dealing with the method of undetermined coefficients for second order linear nonhomogenous differential equations). We can then find a particular solution Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y(t)} if we extend the assumed forms of combinations of polynomials, exponential functions, or sine/cosine functions, and multiplying by powers of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t} to ensure our particular solution does not contain part of a solution to the corresponding higher order linear homogenous differential equation. We can then solve for these constants by plugging our assumed form Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y(t)} into our differential equation above.

Providing a list of possible forms is trivial, so we will instead look at some examples of apply this method.

Example 1

Find the general solution to the differential equation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{d^3y}{dt^3} + \frac{d^2y}{dt^2} + \frac{dy}{dt} + y = e^{-t} + 4t} .

We will need to first solve the corresponding third order linear homogenous differential equation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{d^3y}{dt^3} + \frac{d^2y}{dt^2} + \frac{dy}{dt} + y = 0} . This characteristic equation to this differential equation is:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \quad r^3 + r^2 + r + 1 = 0 \end{align}}

We can (by trial and error) see that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r_1 = -1} is a solution to this characteristic equation. Applying long division with the factor Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (r + 1)} and we have that our characteristic equation can be written as:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \quad (r + 1)(r^2 + 1) = 0 \end{align}}

Therefore we can see that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r_2 = i} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r_3 = -i} . Therefore the general solution to our third order linear homogenous differential equation is:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \quad y_h(t) = C_1e^{-t} + C_2\cos(t) + C_3\sin(t) \end{align}}

Now we note that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g(t) = e^{-t} + 4t} has an exponential term and a cosine term, so we expect the form of our particular solution to be Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y(t) = Ae^{-t} ...} . We now compute the first, second, and third derivatives of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y} .

Licensing

Content obtained and/or adapted from:

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