Solution Of Homogeneous Differential Equations And Examples

Homogeneous Differential Equation:

            Now I want to different forms and types of differential equations.First of all we should focus on the types, that are Homogeneous Differential equations, Exact and non exact differential equations etc. The homogeneous equation must have same degree.

  For example : 

solve 

              x^2y dx - ( x^3+y^3) dy = 0

Solution : 

           The given equation can be written in the form
         dy/ dx = x^2y/x^3+y^3

Putting y= vx, we have

  v+xdv/dx= x^2.vx/x^3+v^3x^3 = v/ 1+v^3

=> x dv/dx = v/1+v^3 - v

                    = -v^4/1+v^3

=>(1+v ³/v⁴)dv = - dx/x

=>(1/v⁴+1/v)dv = - dx/x

Integrating,
     -1/3v³+log(v) = - log(x)+log(c)

=> log(vx/c)=1/3v³

=> vx/c = exp(1/3v³)

=> y = cexp(x³/3y³)  [·.· v= y/x]

Which is the required solution for the above homogeneous differential equation.

     There are some special forms of homogeneous differential equations.Those are
1. dy/dx = ax + by+ c/a'x+b'y+c'


      Where a/a' ≠ b/b'


2. If a/a' = b/b' = 1/λ (say)


I hope it is useful for you.