# Second order Differential equation with non-constant coefficients

When the coefficients are not constant, and one solutions is known, it is easy to use reduction of order to compute the second solution. But what if both solutions are unknown, is there any general approach to the solutions other than guessing one of the solutions? For example: $y''+sin(t)y'+cos(t)y=0$

How could one solve this equation if there is no solution provided ?

Concerning the specific case you give as an example $$y''+\sin(t)\,y'+\cos(t)\,y=0$$ the equation simplifies if we start setting $y=e^z$ which makes the equation to be $$z''+z' \left(z'+\sin (t)\right)+\cos (t)=0$$ for which an "obvious" solution seems to be $z=\cos(t)+c_1$. So $y=c_2\, e^{\cos(t)}$ is a solution.
• If you know one solution $y_0$, then you can find another by setting $y=y_0 z$ and deriving a first-order ODE for $z'$. If I haven't made any mistakes, another solution (linearly independent of the first one) is $y(t)=e^{\cos t} \int_0^t e^{-\cos s} ds$. – Hans Lundmark Mar 25 '18 at 15:09