- #1
K29
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In cullen-zill chapter 6 equation 23 it says that
[itex]y_{2}(x)=y_{1}(x)\int\frac{e^{-\int P(x)dx}}{y_{1}^{2}(x)}dx[/itex]
is a solution of
[itex]y''+P(x)y'+Q(x)y=0[/itex]
whenever [itex]y_{1}(x)[/itex] is a known solution
Where does this come from? I would like to be able to prove this or find a proof somewhere.
My first thought is that since the general solution solution is [itex]y_{h}=C_{1}y_{1}+C_{2}y_{2}[/itex] then
[itex]y_{2}=v(x)y_{1}[/itex] where v(x) is just a function of x.
Thought maybe I could substitute that into the general form of the DE, but it doesn't seem to help much
[itex]y_{2}(x)=y_{1}(x)\int\frac{e^{-\int P(x)dx}}{y_{1}^{2}(x)}dx[/itex]
is a solution of
[itex]y''+P(x)y'+Q(x)y=0[/itex]
whenever [itex]y_{1}(x)[/itex] is a known solution
Where does this come from? I would like to be able to prove this or find a proof somewhere.
My first thought is that since the general solution solution is [itex]y_{h}=C_{1}y_{1}+C_{2}y_{2}[/itex] then
[itex]y_{2}=v(x)y_{1}[/itex] where v(x) is just a function of x.
Thought maybe I could substitute that into the general form of the DE, but it doesn't seem to help much