Homogeneous Differential Equations - Revision history

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A [[differential equation]] can be '''homogeneous''' in either of two respects.

A [[first order differential equation]] is said to be homogeneous if it may be written
:<math>f(x,y) \, dy = g(x,y) \, dx,</math>
where {{mvar|f}} and {{mvar|g}} are [[homogeneous function]]s of the same degree of {{mvar|x}} and {{mvar|y}}.<ref name="Zill2012">{{cite book|author=Dennis G. Zill|title=A First Course in Differential Equations with Modeling Applications|url=https://books.google.com/books?id=pasKAAAAQBAJ&q=homogeneous|date=15 March 2012|publisher=Cengage Learning|isbn=978-1-285-40110-2}}</ref> In this case, the change of variable {{math|1=''y'' = ''ux''}} leads to an equation of the form
:<math>\frac{dx}{x} = h(u) \, du,</math>
which is easy to solve by [[integral|integration]] of the two members.

Otherwise, a differential equation is homogeneous if it is a homogeneous function of the unknown function and its derivatives. In the case of [[linear differential equation]]s, this means that there are no constant terms. The solutions of any linear [[ordinary differential equation]] of any order may be deduced by integration from the solution of the homogeneous equation obtained by removing the constant term.

==History==

The term ''homogeneous'' was first applied to differential equations by [[Johann Bernoulli]] in section 9 of his 1726 article ''De integraionibus aequationum differentialium'' (On the integration of differential equations).<ref>{{cite journal |date=June 1726 |title=De integraionibus aequationum differentialium |url=https://archive.org/details/commentariiacade01impe |journal=Commentarii Academiae Scientiarum Imperialis Petropolitanae |volume=1|pages=167–184}}</ref>

== Homogeneous first-order differential equations ==
{{Differential equations}}

A first-order [[ordinary differential equation]] in the form:

:<math>M(x,y)\,dx + N(x,y)\,dy = 0 </math>

is a homogeneous type if both functions {{math|''M''(''x'', ''y'')}} and {{math|''N''(''x'', ''y'')}} are [[homogeneous function]]s of the same degree {{mvar|n}}.<ref>{{harvnb|Ince|1956|p=18}}</ref> That is, multiplying each variable by a parameter {{math|λ}}, we find

:<math>M(\lambda x, \lambda y) = \lambda^n M(x,y) \quad \text{and} \quad N(\lambda x, \lambda y) = \lambda^n N(x,y)\,. </math>

Thus,
:<math>\frac{M(\lambda x, \lambda y)}{N(\lambda x, \lambda y)} = \frac{M(x,y)}{N(x,y)}\,. </math>

===Solution method===
In the quotient <math display="inline">\frac{M(tx,ty)}{N(tx,ty)} = \frac{M(x,y)}{N(x,y)}</math>, we can let {{math|1=''t'' = {{sfrac|1|''x''}}}} to simplify this quotient to a function {{mvar|f}} of the single variable {{math|{{sfrac|''y''|''x''}}}}:

:<math>\frac{M(x,y)}{N(x,y)} = \frac{M(tx,ty)}{N(tx,ty)} = \frac{M(1,y/x)}{N(1,y/x)}=f(y/x)\,. </math>
That is
:<math>\frac{dy}{dx} = -f(y/x).</math>

Introduce the [[change of variables]] {{math|1=''y'' = ''ux''}}; differentiate using the [[product rule]]:

:<math>\frac{dy}{dx}=\frac{d(ux)}{dx} = x\frac{du}{dx} + u\frac{dx}{dx} = x\frac{du}{dx} + u.</math>

This transforms the original differential equation into the [[Separation of variables|separable]] form
: <math>x\frac{du}{dx} = -f(u) - u, </math>
or
: <math>\frac 1x\frac{dx}{du} = \frac {-1}{f(u) + u}, </math>
which can now be integrated directly: {{math|ln ''x''}} equals the [[antiderivative]] of the right-hand side (see [[ordinary differential equation]]).

===Special case===

A first order differential equation of the form ({{mvar|a}}, {{mvar|b}}, {{mvar|c}}, {{mvar|e}}, {{mvar|f}}, {{mvar|g}} are all constants)
:<math> \left(ax + by + c\right) dx + \left(ex + fy + g\right) dy = 0</math>
where {{math|''af'' ≠ ''be''}}
can be transformed into a homogeneous type by a linear transformation of both variables ({{mvar|α}} and {{mvar|β}} are constants):
:<math>t = x + \alpha; \;\; z = y + \beta \,. </math>

==Homogeneous linear differential equations==
{{see also|Linear differential equation}}
A linear differential equation is '''homogeneous''' if it is a [[homogeneous linear equation]] in the unknown function and its derivatives. It follows that, if {{math|''φ''(''x'')}} is a solution, so is {{math|''cφ''(''x'')}}, for any (non-zero) constant {{mvar|c}}. In order for this condition to hold, each nonzero term of the linear differential equation must depend on the unknown function or any derivative of it. A linear differential equation that fails this condition is called '''inhomogeneous.'''

A [[linear differential equation]] can be represented as a [[linear operator]] acting on {{math|''y''(''x'')}} where {{mvar|x}} is usually the independent variable and {{mvar|y}} is the dependent variable. Therefore, the general form of a [[linear homogeneous differential equation]] is

:<math> L(y) = 0</math>

where {{mvar|L}} is a [[differential operator]], a sum of derivatives (defining the "0th derivative" as the original, non-differentiated function), each multiplied by a function {{math|''f''''i''}} of {{mvar|x}}:

:<math> L = \sum_{i=0}^n f_i(x)\frac{d^i}{dx^i} \, ,</math>
where {{math|''f''''i''}} may be constants, but not all {{math|''f''''i''}} may be zero.

For example, the following linear differential equation is homogeneous:

:<math> \sin(x) \frac{d^2y}{dx^2} + 4 \frac{dy}{dx} + y = 0 \,, </math>

whereas the following two are inhomogeneous:

:<math> 2 x^2 \frac{d^2y}{dx^2} + 4 x \frac{dy}{dx} + y = \cos(x) \,; </math>

:<math> 2 x^2 \frac{d^2y}{dx^2} - 3 x \frac{dy}{dx} + y = 2 \,. </math>
The existence of a constant term is a sufficient condition for an equation to be inhomogeneous, as in the above example.

@@ Line 337: / Line 337: @@
 Content obtained and/or adapted from:
 * [https://en.wikipedia.org/wiki/Homogeneous_differential_equation Homogeneous differential equation, Wikipedia] under a CC BY-SA license
-* [https://en.wikibooks.org/wiki/Ordinary_Differential_Equations/Homogeneous_second_order_equations Homogeneous second order equations, Wikibooks: Ordinary Differential Equations] under a CC BY-SA license
+* [https://en.wikibooks.org/wiki/Ordinary_Differential_Equations/Homogenous_1 Homogeneous 1, Wikibooks: Ordinary Differential Equations] under a CC BY-SA license

← Older revision		Revision as of 15:14, 22 October 2021
Line 117:		Line 117:
	Content obtained and/or adapted from:		Content obtained and/or adapted from:
	* [https://en.wikipedia.org/wiki/Homogeneous_differential_equation Homogeneous differential equation, Wikipedia] under a CC BY-SA license		* [https://en.wikipedia.org/wiki/Homogeneous_differential_equation Homogeneous differential equation, Wikipedia] under a CC BY-SA license
		+	* [https://en.wikibooks.org/wiki/Ordinary_Differential_Equations/Homogeneous_second_order_equations Homogeneous second order equations, Wikibooks: Ordinary Differential Equations] under a CC BY-SA license

@@ Line 72: / Line 72: @@
 The existence of a constant term is a sufficient condition for an equation to be inhomogeneous, as in the above example.
-===Characteristic equation===
+==Characteristic equation==
 If the equation is linear homogeneous and further <math>p(t),q(t)</math> are constant, then the equation is referred to as a '''constant-coefficients''' equation:

@@ Line 71: / Line 71: @@
 :<math> 2 x^2 \frac{d^2y}{dx^2} - 3 x \frac{dy}{dx} + y = 2 \,. </math>
 The existence of a constant term is a sufficient condition for an equation to be inhomogeneous, as in the above example.
 == Licensing ==
 Content obtained and/or adapted from:
 * [https://en.wikipedia.org/wiki/Homogeneous_differential_equation Homogeneous differential equation, Wikipedia] under a CC BY-SA license

@@ Line 1: / Line 1: @@
-A [[differential equation]] can be '''homogeneous''' in either of two respects.
+A differential equation can be '''homogeneous''' in either of two respects.
-A [[first order differential equation]] is said to be homogeneous if it may be written
+A first order differential equation is said to be homogeneous if it may be written
 :<math>f(x,y) \, dy = g(x,y) \, dx,</math>
-where {{mvar|f}} and {{mvar|g}} are [[homogeneous function]]s of the same degree of {{mvar|x}} and {{mvar|y}}.<ref name="Zill2012">{{cite book|author=Dennis G. Zill|title=A First Course in Differential Equations with Modeling Applications|url=https://books.google.com/books?id=pasKAAAAQBAJ&q=homogeneous|date=15 March 2012|publisher=Cengage Learning|isbn=978-1-285-40110-2}}</ref> In this case, the change of variable {{math|1=''y'' = ''ux''}} leads to an equation of the form
+where {{mvar|f}} and {{mvar|g}} are homogeneous functions of the same degree of {{mvar|x}} and {{mvar|y}}. In this case, the change of variable {{math|1=''y'' = ''ux''}} leads to an equation of the form
 :<math>\frac{dx}{x} = h(u) \, du,</math>
-which is easy to solve by [[integral|integration]] of the two members.
+which is easy to solve by integration of the two members.
-Otherwise, a differential equation is homogeneous if it is a homogeneous function of the unknown function and its derivatives. In the case of [[linear differential equation]]s, this means that there are no constant terms. The solutions of any linear [[ordinary differential equation]] of any order may be deduced by integration from the solution of the homogeneous equation obtained by removing the constant term.
+Otherwise, a differential equation is homogeneous if it is a homogeneous function of the unknown function and its derivatives. In the case of linear differential equations, this means that there are no constant terms. The solutions of any linear ordinary differential equation of any order may be deduced by integration from the solution of the homogeneous equation obtained by removing the constant term.
 == Homogeneous first-order differential equations ==
-A first-order [[ordinary differential equation]] in the form:
 :<math>M(x,y)\,dx + N(x,y)\,dy = 0 </math>
-is a homogeneous type if both functions {{math|''M''(''x'', ''y'')}} and {{math|''N''(''x'', ''y'')}} are [[homogeneous function]]s of the same degree {{mvar|n}}.<ref>{{harvnb|Ince|1956|p=18}}</ref> That is, multiplying each variable by a parameter {{math|λ}}, we find
+is a homogeneous type if both functions {{math|''M''(''x'', ''y'')}} and {{math|''N''(''x'', ''y'')}} are homogeneous functions of the same degree {{mvar|n}}. That is, multiplying each variable by a parameter {{math|λ}}, we find
 :<math>M(\lambda x, \lambda y) = \lambda^n M(x,y) \quad \text{and} \quad N(\lambda x, \lambda y) = \lambda^n N(x,y)\,. </math>
@@ Line 24: / Line 24: @@
 ===Solution method===
-In the quotient <math display="inline">\frac{M(tx,ty)}{N(tx,ty)} = \frac{M(x,y)}{N(x,y)}</math>, we can let {{math|1=''t'' = {{sfrac|1|''x''}}}} to simplify this quotient to a function {{mvar|f}} of the single variable {{math|{{sfrac|''y''|''x''}}}}:
+In the quotient <math display="inline">\frac{M(tx,ty)}{N(tx,ty)} = \frac{M(x,y)}{N(x,y)}</math>, we can let <math>t = \frac{1}{x}</math> to simplify this quotient to a function {{mvar|f}} of the single variable <math>\frac{y}{x}</math>:
 :<math>\frac{M(x,y)}{N(x,y)} = \frac{M(tx,ty)}{N(tx,ty)} = \frac{M(1,y/x)}{N(1,y/x)}=f(y/x)\,. </math>
@@ Line 30: / Line 30: @@
 :<math>\frac{dy}{dx} = -f(y/x).</math>
-Introduce the [[change of variables]] {{math|1=''y'' = ''ux''}}; differentiate using the [[product rule]]:
+Introduce the change of variables {{math|1=''y'' = ''ux''}}; differentiate using the product rule:
 :<math>\frac{dy}{dx}=\frac{d(ux)}{dx} = x\frac{du}{dx} + u\frac{dx}{dx} = x\frac{du}{dx} + u.</math>
-This transforms the original differential equation into the [[Separation of variables|separable]] form
+This transforms the original differential equation into the separable form
 : <math>x\frac{du}{dx} = -f(u) - u, </math>
 or
 : <math>\frac 1x\frac{dx}{du} = \frac {-1}{f(u) + u}, </math>
-which can now be integrated directly: {{math|ln ''x''}} equals the [[antiderivative]] of the right-hand side (see [[ordinary differential equation]]).
+which can now be integrated directly: {{math|ln ''x''}} equals the antiderivative of the right-hand side (see ordinary differential equation).
 ===Special case===
@@ Line 49: / Line 49: @@
 ==Homogeneous linear differential equations==
-A [[linear differential equation]] can be represented as a [[linear operator]] acting on {{math|''y''(''x'')}} where {{mvar|x}} is usually the independent variable and {{mvar|y}} is the dependent variable. Therefore, the general form of a [[linear homogeneous differential equation]] is
+A linear differential equation is  '''homogeneous''' if it is a homogeneous linear equation in the unknown function and its derivatives. It follows that, if {{math|''φ''(''x'')}} is a solution, so is {{math|''cφ''(''x'')}}, for any (non-zero) constant {{mvar|c}}. In order for this condition to hold, each nonzero term of the linear differential equation must depend on the unknown function or any derivative of it. A linear differential equation that fails this condition is called '''inhomogeneous.'''
 :<math> L(y) = 0</math>
-where {{mvar|L}} is a [[differential operator]], a sum of derivatives (defining the "0th derivative" as the original, non-differentiated function), each multiplied by a function {{math|''f''<sub>''i''</sub>}} of {{mvar|x}}:
+where {{mvar|L}} is a differential operator, a sum of derivatives (defining the "0th derivative" as the original, non-differentiated function), each multiplied by a function {{math|''f''<sub>''i''</sub>}} of {{mvar|x}}:
 :<math> L = \sum_{i=0}^n f_i(x)\frac{d^i}{dx^i} \, ,</math>
@@ Line 132: / Line 132: @@
 ::<math>y=e^{2x},\ y=e^{-3x}</math>
-as two distinct solutions to the DE. According to the [[../Higher 1#Superposition_principle|superposition principle]], the general solution is therefore
+as two distinct solutions to the DE. According to the superposition principle, the general solution is therefore
 ::<math>y=Ae^{2x}+Be^{-3x} \,</math>