Fundamental Solutions - Revision history

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2021-10-29T16:13:58Z

Fundamental sets of solutions

Lila at 16:12, 29 October 2021

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Lila: /* Fundamental sets of solutions */

2021-10-29T16:09:37Z

Fundamental sets of solutions

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Fundamental sets of solutions

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Lila at 15:55, 29 October 2021

2021-10-29T15:55:16Z

← Older revision		Revision as of 16:15, 29 October 2021
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	Content obtained and/or adapted from:		Content obtained and/or adapted from:
	* [http://mathonline.wikidot.com/fundamental-solutions-to-linear-homogenous-differential-equa Fundamental Solutions to Linear Homogenous Differential Equations, mathonline.wikidot.com] under a CC BY-SA license		* [http://mathonline.wikidot.com/fundamental-solutions-to-linear-homogenous-differential-equa Fundamental Solutions to Linear Homogenous Differential Equations, mathonline.wikidot.com] under a CC BY-SA license
		+	* [http://mathonline.wikidot.com/fundamental-sets-of-solutions-to-a-linear-homogeneous-system Fundamental Sets of Solutions, mathonline.wikidot.com] under a CC BY-SA license

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 <p>For example, consider the following linear homogeneous system of <math>2</math> first order ODEs:</p>
-<math>\begin{align} \quad \left\{\begin{matrix} x_1' = x_1\\ x_2' = 2x_2 \end{matrix}\right. \end{align}</math>
+<div style="text-align: center;"><math>\begin{align} \quad \left\{\begin{matrix} x_1' = x_1\\ x_2' = 2x_2 \end{matrix}\right. \end{align}</math></div>
 <p>We can easily solve this system. For the first differential equation:</p>
-<math>\begin{align} \quad \frac{dx_1}{dt} &= x_1 \\ \frac{dx_1}{x_1} &= dt \\ \int \frac{1}{x_1}  dx &= \int  dt \\ \ln (x_1) &= t + C \\ x_1 &= e^{t + C} \\ x_1 &= C_1e^t \end{align}</math>
+<div style="text-align: center;"><math>\begin{align} \quad \frac{dx_1}{dt} &= x_1 \\ \frac{dx_1}{x_1} &= dt \\ \int \frac{1}{x_1}  dx &= \int  dt \\ \ln (x_1) &= t + C \\ x_1 &= e^{t + C} \\ x_1 &= C_1e^t \end{align}</math></div>
 <p>Where <math>C_1 = e^C > 0</math>.</p>
 <p>For the second differential equation:</p>
-<math>\begin{align} \quad \frac{dx_2}{dt} &= 2x_2 \\ \frac{dx_2}{x_2} &= 2  dt \\ \int \frac{1}{x_2}  dx_2 &= \int 2  dt \\ \ln (x_2) &= 2t + C \\ x_2 &= e^{2t + C} \\ x_2 &= C_2e^{2t} \end{align}</math>
+<div style="text-align: center;"><math>\begin{align} \quad \frac{dx_2}{dt} &= 2x_2 \\ \frac{dx_2}{x_2} &= 2  dt \\ \int \frac{1}{x_2}  dx_2 &= \int 2  dt \\ \ln (x_2) &= 2t + C \\ x_2 &= e^{2t + C} \\ x_2 &= C_2e^{2t} \end{align}</math></div>
 <p>Where <math>C_2 = e^C > 0</math>.</p>
 <p>Note that in fact <math>C_1, C_2</math> can be any real numbers are we are not simply restricted to <math>C_1, C_2 > 0</math>.</p>
 <p>Now by taking <math>C_1 = 1</math>, <math>C_2 = 0</math> we get that <math>\phi^{[1]} = \begin{bmatrix} e^t\\ 0 \end{bmatrix}</math> is a solution to this system. Also, by taking <math>C_1 = 0</math> and <math>C_2 = 1</math> we get that <math>\phi^{[2]} = \phi^{[1]} = \begin{bmatrix} 0\\ e^{2t} \end{bmatrix}</math> is a solution to this system.</p>
 <p>We will now show that <math>\{ \phi^{[1]}, \phi^{[2]} \}</math> is a Fundamental set of solutions to this system on all of <math>\mathbb{R}</math>. Let <math>\alpha, beta \in \mathbb{R}</math> and consider the following equation:</p>
-<math>\begin{align} \alpha \phi^{[1]} + \beta \phi^{[2]} &= 0 \\ \alpha \begin{bmatrix} e^t\\ 0 \end{bmatrix} + \beta \begin{bmatrix} 0\\ e^{2t} \end{bmatrix} &= \begin{bmatrix} 0 \\ 0 \end{bmatrix} \\ \begin{bmatrix} \alpha e^t \\ \beta e^{2t} \end{bmatrix} &= \begin{bmatrix} 0 \\ 0 \end{bmatrix} \end{align}</math>
+<div style="text-align: center;"><math>\begin{align} \alpha \phi^{[1]} + \beta \phi^{[2]} &= 0 \\ \alpha \begin{bmatrix} e^t\\ 0 \end{bmatrix} + \beta \begin{bmatrix} 0\\ e^{2t} \end{bmatrix} &= \begin{bmatrix} 0 \\ 0 \end{bmatrix} \\ \begin{bmatrix} \alpha e^t \\ \beta e^{2t} \end{bmatrix} &= \begin{bmatrix} 0 \\ 0 \end{bmatrix} \end{align}</math></div>
 <p>The equation above implies that <math>\alpha e^t = 0</math> and <math>\beta e^{2t} = 0</math> for all <math>t \in \mathbb{R}</math>. Since <math>e^t, e^{2t} > 0</math> for all <math>t \in \mathbb{R}</math> this implies that <math>\alpha, \beta = 0</math>. So <math>\{ \phi^{[1]}, \phi^{[2]} \}</math> is a linearly independent set of solutions to this system and so <math>\{ \phi^{[1]}, \phi^{[2]} \}</math> is a fundamental set of solutions to this system on <math>\mathbb{R}</math>.</p>

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 </ul>
-<math>\begin{align} \phi(t) = Cy_1(t) + Dy_2(t) \end{align}</math>
+<div style="text-align: center;"><math>\begin{align} \phi(t) = Cy_1(t) + Dy_2(t) \end{align}</math></div>
 <ul>
 <li>So all solutions for this differential equation are a linear combination of the solutions <math>y = y_1(t)</math> and <math>y = y_2(t)</math>.</li>
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 <p>Theorem 1 above implies that if we can find two solutions <math>y = y_1(t)</math> and <math>y = y_2(t)</math> for which the Wronskian <math>W(y_1, y_2) \neq 0</math>, then for constants <math>C</math> and <math>D</math>, all solutions of the second order linear homogenous differential equation <math>\frac{d^2 y}{dt^2} + p(t) \frac{dy}{dt} + q(t) y = 0</math> are given by:</p>
-<math>\begin{align} \quad y = Cy_1(t) + Dy_2(t) \end{align}</math>
+<div style="text-align: center;"><math>\begin{align} \quad y = Cy_1(t) + Dy_2(t) \end{align}</math></div>
 <p>Also note that thus far we have not said that <math>y = y_1(t)</math> and <math>y = y_2(t)</math> need to be distinct. However, with the Theorem above, we see that if <math>y_1(t) = y_2(t)</math> then the Wronskian <math>W(y_1, y_2) = W(y_1, y_1) = W(y_2, y_2)</math> is zero (as you should verify) and so not all solutions to a second order linear homogenous differential are given by the linear combination of just <math>y_1</math>.</p>
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 </ul>
-<math>\begin{align} \quad W(y_1, y_2) \bigg|_{t_0} = \begin{vmatrix} y_1(t_0)  y_2(t_0) \\ y_1'(t_0)  y_2'(t_0)\end{vmatrix} = \begin{vmatrix} 1  0\\ 0  1 \end{vmatrix} = 1 \neq 0 \end{align}</math>
+<div style="text-align: center;"><math>\begin{align} \quad W(y_1, y_2) \bigg|_{t_0} = \begin{vmatrix} y_1(t_0)  y_2(t_0) \\ y_1'(t_0)  y_2'(t_0)\end{vmatrix} = \begin{vmatrix} 1  0\\ 0  1 \end{vmatrix} = 1 \neq 0 \end{align}</math></div>
 <ul>
 <li>Thus Theorem 1 implies that ALL solutions to this differential equation are given by <math>y = Cy_1(t) + Dy_2(t)</math> where <math>C</math> and <math>D</math> are constants. Thus <math>y_1</math> and <math>y_2</math> form a fundamental set of solutions for this differential equation. <math>\blacksquare</math></li>

@@ Line 11: / Line 11: @@
 </ul>
-<math>\begin{align} \quad W(y_1, y_2) \bigg|_{t_0} \neq 0 \end{align}</math>
+<div style="text-align: center;"><math>\begin{align} \quad W(y_1, y_2) \bigg|_{t_0} \neq 0 \end{align}</math></div>
 <ul>
 <li>Take this value <math>t_0</math> and evaluate both <math>\phi</math> and <math>\phi'</math> at this point. Then <math>\phi (t_0) = y_0</math> and <math>\phi'(t_0) = y'_0</math> (since <math>y = \phi(t)</math> is a solution to our differential equation). Now consider the initial value problem <math>\frac{d^2 y}{dt^2} + p(t) \frac{dy}{dt} + q(t) y = 0</math> with the initial conditions <math>y(t_0) = y_0</math> and <math>y'(t_0) = y'_0</math>. The function <math>\phi</math> satisfies this differential equation. Since <math>W(y_1, y_2) \bigg|_{t_0} \neq 0</math> then we have that there exists constants <math>C</math> and <math>D</math> such that <math>y = Cy_1(t) + Dy_2(t)</math> satisfies this initial value problem. But since <math>p</math> and <math>q</math> are continuous on the open interval <math>I</math> containing <math>t_0</math> then this implies that a unique solution exists, and so:</li>

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 ==Fundamental sets of solutions==
+<blockquote style="background: white; border: 1px solid black; padding: 1em;">
-<td><strong>Definition:</strong> A <strong>Fundamental Set of Solutions</strong> to the linear homogeneous system of first order ODEs <math>\mathbf{x}' = A(t) \mathbf{x}</math> on <math>J = (a, b)</math> is a set <math>\{ \phi^{[1]}, \phi^{[2]}, ..., \phi^{[n]} \}</math> of linearly independent solutions to this system on <math>J</math>.</td>
+<td><strong>Definition:</strong> A <strong>Fundamental Set of Solutions</strong> to the linear homogeneous system of first order ODEs <math>\mathbf{x}' = A(t) \mathbf{x}</math> on <math>J = (a, b)</math> is a set <math>\{ \phi^{[1]}, \phi^{[2]}, ..., \phi^{[n]} \}</math> of linearly independent solutions to this system on <math>J</math>.</td></blockquote>
 <p>For example, consider the following linear homogeneous system of <math>2</math> first order ODEs:</p>

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 <blockquote style="background: white; border: 1px solid black; padding: 1em;">
 <td><strong>Theorem 1:</strong> Let <math>\frac{d^2 y}{dt^2} + p(t) \frac{dy}{dt} + q(t)y = 0</math> be a second order linear homogenous differential equation where <math>p</math> and <math>q</math> are continuous on an open interval <math>I</math> such that <math>t_0 \in I</math>, and let <math>y = y_1(t)</math> and <math>y = y_2(t)</math> be two solutions to this differential equation. The set of all linear combinations of these two solutions, <math>y = Cy_1(t) + Dy_2(t)</math> where <math>C</math> and <math>D</math> are constants contains all solutions to this differential equation if and only if there exists a point <math>t_0</math> for which the Wronksian of <math>y_1</math> and <math>y_2</math> at <math>t_0</math> is nonzero, that is <math>W(y_1, y_2) \bigg|_{t_0} \neq 0</math>.</td></blockquote>
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 <li>Thus Theorem 1 implies that ALL solutions to this differential equation are given by <math>y = Cy_1(t) + Dy_2(t)</math> where <math>C</math> and <math>D</math> are constants. Thus <math>y_1</math> and <math>y_2</math> form a fundamental set of solutions for this differential equation. <math>\blacksquare</math></li>
 </ul>
 ==Resources==

← Older revision		Revision as of 15:59, 29 October 2021
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	Content obtained and/or adapted from:		Content obtained and/or adapted from:
	* [http://mathonline.wikidot.com/fundamental-solutions-to-linear-homogenous-differential-equa Fundamental Solutions to Linear Homogenous Differential Equations, mathonline.wikidot.com] under a CC BY-SA license		* [http://mathonline.wikidot.com/fundamental-solutions-to-linear-homogenous-differential-equa Fundamental Solutions to Linear Homogenous Differential Equations, mathonline.wikidot.com] under a CC BY-SA license
−
−	~~<math>\begin{align} \quad \frac{dx_1}{dt} & = x_1 \\ \frac{dx_1}{x_1} & = dt \\ \int \frac{1}{x_1} dx & = \int dt \\ \ln (x_1) & = t + C \\ x_1 & = e^{t + C} \\ x_1 & = C_1e^t \end{align}</math>~~

← Older revision		Revision as of 15:56, 29 October 2021
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	* [http://mathonline.wikidot.com/fundamental-solutions-to-linear-homogenous-differential-equa Fundamental Solutions to Linear Homogenous Differential Equations, mathonline.wikidot.com] under a CC BY-SA license		* [http://mathonline.wikidot.com/fundamental-solutions-to-linear-homogenous-differential-equa Fundamental Solutions to Linear Homogenous Differential Equations, mathonline.wikidot.com] under a CC BY-SA license

−	<math>\begin{align} \quad \frac{dx_1}{dt} &~~amp;~~= x_1 \\ \frac{dx_1}{x_1} &~~amp;~~= dt \\ \int \frac{1}{x_1} \: dx &~~amp;~~= \int \: dt \\ \ln (x_1) &~~amp;~~= t + C \\ x_1 &~~amp;~~= e^{t + C} \\ x_1 &~~amp;~~= C_1e^t \end{align}</math>	+	<math>\begin{align} \quad \frac{dx_1}{dt} & = x_1 \\ \frac{dx_1}{x_1} & = dt \\ \int \frac{1}{x_1} dx & = \int dt \\ \ln (x_1) & = t + C \\ x_1 & = e^{t + C} \\ x_1 & = C_1e^t \end{align}</math>

← Older revision		Revision as of 15:55, 29 October 2021
Line 53:		Line 53:
	Content obtained and/or adapted from:		Content obtained and/or adapted from:
	* [http://mathonline.wikidot.com/fundamental-solutions-to-linear-homogenous-differential-equa Fundamental Solutions to Linear Homogenous Differential Equations, mathonline.wikidot.com] under a CC BY-SA license		* [http://mathonline.wikidot.com/fundamental-solutions-to-linear-homogenous-differential-equa Fundamental Solutions to Linear Homogenous Differential Equations, mathonline.wikidot.com] under a CC BY-SA license
		+
		+	<math>\begin{align} \quad \frac{dx_1}{dt} &= x_1 \\ \frac{dx_1}{x_1} &= dt \\ \int \frac{1}{x_1} \: dx &= \int \: dt \\ \ln (x_1) &= t + C \\ x_1 &= e^{t + C} \\ x_1 &= C_1e^t \end{align}</math>