I have been trying to solve this problem for a while. However, I am not sure on how to even get started. Any hint will help.
By completing the following steps, show that the general solution of a second order homogeneous linear equation $ay'' + by' + cy = 0$ is of the form $c_1y_1(x) +c_2y_2(x)$ where $y_1(x)$ and $y_2(x)$ are linearly independent solutions.
(a) Show that two functions $f(x)$ and $g(x)$ are linearly dependent on some open interval $I$ if and only if their Wronskian function $W[f; g](x)$, defined as $f(x)g'(x) ̶ f'(x)g(x)$ is zero for all $x \in I$.
(b) Show that if $y(x)$ and $z(x)$ are any solutions of $ay'' + by' + cy = 0$, then $W[y; z](x)$ is a solution of $aW'(x) + bW(x) = 0$. Thus $W[y; z](x) = C \exp (-bx=a)$ for some constant $C$ which depends on the choice of solutions $y$ and $z$. (This is Abel's formula.)
(c) Conclude from (b) that if $W[y; z](x) = 0$ for some $x$, then $W[y; z](x) = 0$ for all $x$.