Homogeneous differential equation

<h2 id="history">History</h2>
The term homogeneous was first applied to differential equations by <a href="/facts/Johann_Bernoulli/U8kRr2Bn">Johann Bernoulli</a> in section 9 of his 1726 article De integraionibus aequationum differentialium (On the integration of differential equations).<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a>

<h2 id="homogeneous-first-order-differential-equations">Homogeneous first-order differential equations</h2>

A first-order <a href="/facts/Ordinary_differential_equation/ygKGQ8kD">ordinary differential equation</a> in the form:

M
        (
        x
        ,
        y
        )
        
        d
        x
        +
        N
        (
        x
        ,
        y
        )
        
        d
        y
        =
        0
      
    
    {\displaystyle M(x,y)\,dx+N(x,y)\,dy=0}

is a homogeneous type if both functions M(x, y) and N(x, y) are <a href="/facts/Homogeneous_function/wF9XX7s5">homogeneous functions</a> of the same degree n.<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a> That is, multiplying each variable by a parameter λ, we find

M
        (
        λ
        x
        ,
        λ
        y
        )
        =
        
          λ
          
            n
          
        
        M
        (
        x
        ,
        y
        )
        
        
          and
        
        
        N
        (
        λ
        x
        ,
        λ
        y
        )
        =
        
          λ
          
            n
          
        
        N
        (
        x
        ,
        y
        )
        
        .
      
    
    {\displaystyle M(\lambda x,\lambda y)=\lambda ^{n}M(x,y)\quad {\text{and}}\quad N(\lambda x,\lambda y)=\lambda ^{n}N(x,y)\,.}

Thus,

M
              (
              λ
              x
              ,
              λ
              y
              )
            
            
              N
              (
              λ
              x
              ,
              λ
              y
              )
            
          
        
        =
        
          
            
              M
              (
              x
              ,
              y
              )
            
            
              N
              (
              x
              ,
              y
              )
            
          
        
        
        .
      
    
    {\displaystyle {\frac {M(\lambda x,\lambda y)}{N(\lambda x,\lambda y)}}={\frac {M(x,y)}{N(x,y)}}\,.}

<h3>Solution method</h3>
In the quotient 
 
 
 
 
 
 
 M
 (
 t
 x
 ,
 t
 y
 )
 
 
 N
 (
 t
 x
 ,
 t
 y
 )
 
 
 
 =
 
 
 
 M
 (
 x
 ,
 y
 )
 
 
 N
 (
 x
 ,
 y
 )
 
 
 
 
 
 {\textstyle {\frac {M(tx,ty)}{N(tx,ty)}}={\frac {M(x,y)}{N(x,y)}}}
 
, we can let t = ⁠1/x⁠ to simplify this quotient to a function f of the single variable ⁠y/x⁠:

M
              (
              x
              ,
              y
              )
            
            
              N
              (
              x
              ,
              y
              )
            
          
        
        =
        
          
            
              M
              (
              t
              x
              ,
              t
              y
              )
            
            
              N
              (
              t
              x
              ,
              t
              y
              )
            
          
        
        =
        
          
            
              M
              (
              1
              ,
              y
              
                /
              
              x
              )
            
            
              N
              (
              1
              ,
              y
              
                /
              
              x
              )
            
          
        
        =
        f
        (
        y
        
          /
        
        x
        )
        
        .
      
    
    {\displaystyle {\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)\,.}

That is

d
              y
            
            
              d
              x
            
          
        
        =
        −
        f
        (
        y
        
          /
        
        x
        )
        .
      
    
    {\displaystyle {\frac {dy}{dx}}=-f(y/x).}

Introduce the <a href="/facts/Change_of_variables/vjWdXCDF">change of variables</a> y = ux; differentiate using the <a href="/facts/Product_rule/MyKxsP5t">product rule</a>:

d
              y
            
            
              d
              x
            
          
        
        =
        
          
            
              d
              (
              u
              x
              )
            
            
              d
              x
            
          
        
        =
        x
        
          
            
              d
              u
            
            
              d
              x
            
          
        
        +
        u
        
          
            
              d
              x
            
            
              d
              x
            
          
        
        =
        x
        
          
            
              d
              u
            
            
              d
              x
            
          
        
        +
        u
        .
      
    
    {\displaystyle {\frac {dy}{dx}}={\frac {d(ux)}{dx}}=x{\frac {du}{dx}}+u{\frac {dx}{dx}}=x{\frac {du}{dx}}+u.}

This transforms the original differential equation into the <a href="/facts/Separation_of_variables/oJdkw5xq">separable</a> form

x
        
          
            
              d
              u
            
            
              d
              x
            
          
        
        =
        −
        f
        (
        u
        )
        −
        u
        ,
      
    
    {\displaystyle x{\frac {du}{dx}}=-f(u)-u,}

or

1
            x
          
        
        
          
            
              d
              x
            
            
              d
              u
            
          
        
        =
        
          
            
              −
              1
            
            
              f
              (
              u
              )
              +
              u
            
          
        
        ,
      
    
    {\displaystyle {\frac {1}{x}}{\frac {dx}{du}}={\frac {-1}{f(u)+u}},}

which can now be integrated directly: ln x equals the <a href="/facts/Antiderivative/5qDsudIv">antiderivative</a> of the right-hand side (see <a href="/facts/Ordinary_differential_equation/ygKGQ8kD">ordinary differential equation</a>).

<h3>Special case</h3>
A first order differential equation of the form (a, b, c, e, f, g are all constants)

(
          
            a
            x
            +
            b
            y
            +
            c
          
          )
        
        d
        x
        +
        
          (
          
            e
            x
            +
            f
            y
            +
            g
          
          )
        
        d
        y
        =
        0
      
    
    {\displaystyle \left(ax+by+c\right)dx+\left(ex+fy+g\right)dy=0}

where af ≠ be
can be transformed into a homogeneous type by a linear transformation of both variables (α and β are constants):

t
        =
        x
        +
        α
        ;
        
        
        z
        =
        y
        +
        β
        
        ,
      
    
    {\displaystyle t=x+\alpha ;\;\;z=y+\beta \,,}

where

α
        =
        
          
            
              c
              f
              −
              b
              g
            
            
              a
              f
              −
              b
              e
            
          
        
        ;
        
        
        β
        =
        
          
            
              a
              g
              −
              c
              e
            
            
              a
              f
              −
              b
              e
            
          
        
        
        .
      
    
    {\displaystyle \alpha ={\frac {cf-bg}{af-be}};\;\;\beta ={\frac {ag-ce}{af-be}}\,.}

For cases where af = be, introduce the <a href="/facts/Change_of_variables/vjWdXCDF">change of variables</a> u = ax + by or u = ex + fy; differentiation yields:

d
              u
            
            
              d
              x
            
          
        
        =
        a
        −
        b
        
          
            (
          
        
        
          
            
              a
              c
              +
              a
              u
            
            
              a
              g
              +
              e
              u
            
          
        
        
          
            )
          
        
        ,
      
    
    {\displaystyle {\frac {du}{dx}}=a-b{\bigg (}{\frac {ac+au}{ag+eu}}{\bigg )},}

or

d
              u
            
            
              d
              x
            
          
        
        =
        e
        −
        f
        
          
            (
          
        
        
          
            
              e
              c
              +
              a
              u
            
            
              e
              g
              +
              e
              u
            
          
        
        
          
            )
          
        
        ,
      
    
    {\displaystyle {\frac {du}{dx}}=e-f{\bigg (}{\frac {ec+au}{eg+eu}}{\bigg )},}

for each respective substitution. Both may be solved via <a href="/facts/Separation_of_Variables/oJdkw5xq">Separation of Variables</a>.

<h2 id="homogeneous-linear-differential-equations">Homogeneous linear differential equations</h2>
See also: <a href="/facts/Linear_differential_equation/nLEbIIVf">Linear differential equation</a>
A linear differential equation is homogeneous if it is a <a href="/facts/Homogeneous_linear_equation/fk9CFdgp">homogeneous linear equation</a> in the unknown function and its derivatives. It follows that, if φ(x) is a solution, so is cφ(x), for any (non-zero) constant 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 <a href="/facts/Linear_differential_equation/nLEbIIVf">linear differential equation</a> can be represented as a <a href="/facts/Linear_operator/Q1PBKNVV">linear operator</a> acting on y(x) where x is usually the independent variable and y is the dependent variable. Therefore, the general form of a <a href="/facts/Linear_homogeneous_differential_equation/nLEbIIVf">linear homogeneous differential equation</a> is

L
        (
        y
        )
        =
        0
      
    
    {\displaystyle L(y)=0}

where L is a <a href="/facts/Differential_operator/Nr15J0IE">differential operator</a>, a sum of derivatives (defining the "0th derivative" as the original, non-differentiated function), each multiplied by a function fi of x:

L
        =
        
          ∑
          
            i
            =
            0
          
          
            n
          
        
        
          f
          
            i
          
        
        (
        x
        )
        
          
            
              d
              
                i
              
            
            
              d
              
                x
                
                  i
                
              
            
          
        
        
        ,
      
    
    {\displaystyle L=\sum _{i=0}^{n}f_{i}(x){\frac {d^{i}}{dx^{i}}}\,,}

where fi may be constants, but not all fi may be zero.
For example, the following linear differential equation is homogeneous:

sin
        ⁡
        (
        x
        )
        
          
            
              
                d
                
                  2
                
              
              y
            
            
              d
              
                x
                
                  2
                
              
            
          
        
        +
        4
        
          
            
              d
              y
            
            
              d
              x
            
          
        
        +
        y
        =
        0
        
        ,
      
    
    {\displaystyle \sin(x){\frac {d^{2}y}{dx^{2}}}+4{\frac {dy}{dx}}+y=0\,,}

whereas the following two are inhomogeneous:

2
        
          x
          
            2
          
        
        
          
            
              
                d
                
                  2
                
              
              y
            
            
              d
              
                x
                
                  2
                
              
            
          
        
        +
        4
        x
        
          
            
              d
              y
            
            
              d
              x
            
          
        
        +
        y
        =
        cos
        ⁡
        (
        x
        )
        
        ;
      
    
    {\displaystyle 2x^{2}{\frac {d^{2}y}{dx^{2}}}+4x{\frac {dy}{dx}}+y=\cos(x)\,;}

2
        
          x
          
            2
          
        
        
          
            
              
                d
                
                  2
                
              
              y
            
            
              d
              
                x
                
                  2
                
              
            
          
        
        −
        3
        x
        
          
            
              d
              y
            
            
              d
              x
            
          
        
        +
        y
        =
        2
        
        .
      
    
    {\displaystyle 2x^{2}{\frac {d^{2}y}{dx^{2}}}-3x{\frac {dy}{dx}}+y=2\,.}

The existence of a constant term is a sufficient condition for an equation to be inhomogeneous, as in the above example.

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Separation_of_variables/oJdkw5xq">Separation of variables</a></li></ul>
<h2 id="notes">Notes</h2>

<ul><li>Boyce, William E.; DiPrima, Richard C. (2012), Elementary differential equations and boundary value problems (10th ed.), Wiley, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0470458310. (This is a good introductory reference on differential equations.)</li>
<li>Ince, E. L. (1956), <a href="https://archive.org/details/ordinarydifferen029666mbp">Ordinary differential equations</a>, New York: Dover Publications, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0486603490. (This is a classic reference on ODEs, first published in 1926.)</li>
<li>Andrei D. Polyanin; Valentin F. Zaitsev (15 November 2017). <a href="https://books.google.com/books?id=L3JQDwAAQBAJ&q=homogeneous">Handbook of Ordinary Differential Equations: Exact Solutions, Methods, and Problems</a>. CRC Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-4665-6940-9.</li>
<li>Matthew R. Boelkins; Jack L. Goldberg; Merle C. Potter (5 November 2009). <a href="https://books.google.com/books?id=s1-mZMg5fLgC&q=homogeneous&pg=PA274">Differential Equations with Linear Algebra</a>. Oxford University Press. pp. 274–. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-19-973666-9.</li></ul>
<h2 id="external-links">External links</h2>
<ul><li><a href="http://mathworld.wolfram.com/HomogeneousOrdinaryDifferentialEquation.html">Homogeneous differential equations at MathWorld</a></li>
<li><a href="https://en.wikibooks.org/wiki/Ordinary_Differential_Equations/Substitution_1">Wikibooks: Ordinary Differential Equations/Substitution 1</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Dennis G. Zill (15 March 2012). A First Course in Differential Equations with Modeling Applications. Cengage Learning. ISBN 978-1-285-40110-2. <a href="978-1-285-40110-2" target="_blank">978-1-285-40110-2</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">"De integraionibus aequationum differentialium". Commentarii Academiae Scientiarum Imperialis Petropolitanae. 1: 167–184. June 1726. <a href="https://archive.org/details/commentariiacade01impe" target="_blank">https://archive.org/details/commentariiacade01impe</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Ince 1956, p. 18 - Ince, E. L. (1956), Ordinary differential equations, New York: Dover Publications, ISBN 0486603490 <a href="https://archive.org/details/ordinarydifferen029666mbp" target="_blank">https://archive.org/details/ordinarydifferen029666mbp</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
</ol>

Homogeneous differential equation open-in-new

Homogeneous differential equation