I have the diff. eq. $$y''+p\left(x\right)y'+q\left(x\right)y=f\left(x\right)\:$$ with two solutions $y_1\left(x\right),\:y_2\left(x\right)$.
What are the linear combinations of $y_1\left(x\right),\:y_2\left(x\right)$, i.e: $c_1y_1+c_2y_2$ which are also solutions to the diff. eq?
From plugging-in the expression $c_1y_1+c_2y_2$ and from: $y_1''+p\left(x\right)y_1'+q\left(x\right)y_1=f\left(x\right)\:$, $y_2''+p\left(x\right)y_2'+q\left(x\right)y_2=f\left(x\right)\:$
I can get: $$f\left(x\right)\left(c_1+c_2\right)=f\left(x\right)\:$$ which implies: $$c_1=1-c_2$$ Thus: $$\left(1-c_2\right)y_1+c_2y_2$$ is a valid solution for the diff. eq.
Can I say something more then that about the solutions which are a linear combinations of $y_1, y_2$ ?