Solving ODE for $\sigma$: Transformation and Manipulation?

[Juggler123]

Hi all,

I have an ODE of the form

\frac{d^{3}\psi}{d\xi^{3}}-A\left(\psi+\xi\frac{d\psi}{d\xi}\right)=0,

where \psi=C_{1}U(\xi)+C_{2}V(\xi).

Is there any transformation or inventive manipulation I can use here to obtain an ODE for \sigma=U(\xi)+V(\xi)? As this is the quantity I would like to solve for.

Thanks.

 

[JJacquelin]

Hi !
y''(x)-A(y(x)+x*y'(x))=0
Let t=A*x²/2
Then a first solution is easy to see :
U=exp(t)=exp(A*x²/2)
Let y(x)=f(x)*exp(A*x²/2) and z=x*sqrt(A/2)
leading to an ODE which a solution is erf(z)
V= erf(x*sqrt(A/2))*exp(A*x²/2)