How do I choose s(x) when transforming a second-order ODE?

[gankutsuou7]

Every second order linear ODE a(x)y''+b(x)y'+c(x)y=F(x) can be transformed into the forms;

u''+g(x)u=h(x)

divide whole equation to a(x) and choose y(x)=u(x)^(e^s(x)) but how i choose s(x) ?Please help...

 

[jackmell]

If you mean $y=ue^{s(x)}$, then that's a standard transformation to it's normal form. Write:

$$y''+Py'+Qy=g$$

then let:

$$y=ue^{s(x)}$$

with:

$$s(x)=exp\left\{-1/2\int P dx\right\}$$

See "Intermediate Differential Equations" by Rainville