Help me to solve this non-homogeneous differential equation :

$ y''+y=\tan x $

$ 0<x<\dfrac{\pi}{2} $

I could reach to $y_{c}=c_{1}\cos x + c_{2}\sin x$ but particular solution is where I stopped.

  • $\begingroup$ this will help you $\endgroup$ – Dr. Sonnhard Graubner May 29 '15 at 18:31
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    $\begingroup$ I think you mean "particular solution" :) What methods do you have for nonhomogeneous ODEs? Do you know variation of parameters? $\endgroup$ – GPerez May 29 '15 at 18:52
  • $\begingroup$ @GPerez: excuse me because of my mistake...yes I know variation of parameter.in fact my problem is solving $u1=\int{\dfrac{-y_{2}*f(x)}{w(y_{1},y_{2})}}$ I also found $w(y_{1},y_{2})=1$ but something doesn't right when I move forward... $\endgroup$ – sajjad May 29 '15 at 19:02

Knowing a solution of the homogeneous ODE, for example $c\cos(x)$, remplace the constant $c$ by a function $Y(x)$ , i.e.: change of function $y=Y(x)\cos(x)$. This allows to reduce the second order ODE to a first order ODE easier to solve.

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