Ordinary differential equation- solutions of differential equation

Ordinary differential equation- solutions of differential equation

If $¦Õ_1 (x)$ and $¦Õ_2 (x)$ are the solutions of $y''+q(x)y(x)=0$,
$0¡Üx<¡Þ$ such that $W(¦Õ_1,¦Õ_2 )(x)= 1$ and $¦×(x)$ is any solution of
$y''+(q(x)+r(x) )y(x)=0$, $0¡Üx<¡Þ$, there are unique constants $c_1$ and
$c_2$ such that $$¦×(x)= c_1 ¦Õ_1 (x)+ c_2 ¦Õ_2 (x)+ ¡Ò_a^x(¦Õ_1 (x) ¦Õ_2
(t)-¦Õ_2 (x)¦Õ_1 (t)) r(t)¦×(t)dt,$$where $a$ is any point in $[0,¡Þ)$.
Conversely for any constant $c_1$ and $c_2$,there is a unique solution
$¦×(x)$ of $y''+(q(x)+r(x) )y(x)=0$, $0¡Üx<¡Þ$ such that $$¦×(x)= c_1 ¦Õ_1
(x)+ c_2 ¦Õ_2 (x)+ ¡Ò_a^x (¦Õ_1 (x) ¦Õ_2 (t)-¦Õ_2 (x)¦Õ_1 (t))
r(t)¦×(t)dt$$ holds.