# ode/pde: Show that two functions f(x) and g(x) are linearly dependent on some open interval I

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 $x$.

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For (a), rearrange $fg' - f'g = 0$ by putting anything to do with $g$ on the left and anything to do with $f$ on the right. This equation will involve derivatives, but you want to relate $f$ and $g$ (in particular, show they're linearly dependent) not their derivatives, so try to undo the differentiation. – Michael Albanese Jan 17 '13 at 0:29
Statement $(a)$ is not true. – Artem Jan 17 '13 at 3:28