Making an equation self adjoint: P(x)y" + Q(x)y' +R(x)=

[Guest]

I'm having a bit of difficulty showing that any equation P(x)y" + Q(x)y' +R(x)=0 can made self adjoint by multiplying through by:

(1/P)e^integral(Q/P)dx

Hope someone can help me out or point me in the right direction.

thanks

 

[galactus]

I found this in an old DiffEq text. Hope it helps. Different variables than you are using, but just change them.

\(\displaystyle \L\\y"e^{\int{(\frac{b}{a})}dx}+\frac{b(x)}{a(x)}y'e^{\int{(\frac{b}{a})}dx}+\left(\frac{c(x)}{a(x)}e^{\int{(\frac{b}{a})}dx}+{\lambda}\frac{d(x)}{a(x)}e^{\int{(\frac{b}{a})}dx}\right)=0\)


is the same as:

\(\displaystyle \L\\\frac{d}{dx}\left[y'\overbrace{e^{\int{(\frac{b}{a})}dx}}^{\text{r(x)}}\right]+\left(\overbrace{\frac{c(x)}{a(x)}e^{\int{(\frac{b}{a})}dx}}^{\text{q(x)}}+{\lambda}\overbrace{\frac{d(x)}{a(x)}e^{\int{(\frac{b}{a})}dx}}^{\text{p(x)}}\right)y=0\)

I hope this helps.