Does Reduction of Order Always Solve Non-Homogeneous Second Order ODEs?

[Benny]

Hi, say I have a second order ODE $$a_1 \left( x \right)\frac{{d^2 y}}{{dx}} + a_2 \left( x \right)\frac{{dy}}{{dx}} + a_3 \left( x \right)y = f\left( x \right)$$ and I have $$y_H = h\left( x \right)$$ which satisfies the DE for case of f(x) = 0. Can I always, in principle, find the general solution to the DE for the case where f(x) is not necessarily zero by using the substitution $$y\left( x \right) = u\left( x \right)h\left( x \right)$$. I know that there are various substitutions for second order ODEs but I would like to know if the one I mentioned always works. Any help would be good thanks.

 

[saltydog]

Hi, say I have a second order ODE $$a_1 \left( x \right)\frac{{d^2 y}}{{dx}} + a_2 \left( x \right)\frac{{dy}}{{dx}} + a_3 \left( x \right)y = f\left( x \right)$$ and I have $$y_H = h\left( x \right)$$ which satisfies the DE for case of f(x) = 0. Can I always, in principle, find the general solution to the DE for the case where f(x) is not necessarily zero by using the substitution $$y\left( x \right) = u\left( x \right)h\left( x \right)$$. I know that there are various substitutions for second order ODEs but I would like to know if the one I mentioned always works. Any help would be good thanks.

Absolutely. The problem of course is effecting the integration: After letting y=uh and substituting into the ODE, then let:

$$w=u^{'}$$

resulting in:

$$w^{'}+\left(\frac{2h^{'}}{h}+\frac{a_2}{a_1}\right)w=\frac{f}{a_1h}$$