[Math] Show that the Wronskian of solutions of $y”+p(x)y’+q(x)y=0$ satisfies $\frac{dW}{dx}+pW=0$

Question

So I am given: $\{y_1(x),y_2(x)\}$ is a fundamental solution set of the ODE:
$$y''+p(x)y'+q(x)y=0$$ I need to show that the Wronskian $W(y_1,y_2)$ satisfies the ODE $\frac{dW}{dx}+pW=0$ and hence, $W(x)=C \cdot \exp(-\int p(x) dx)$

I calculated the Wronskian to be $W=y_1y_2'-y_1'y_2$, then $\frac{dW}{dx}=y_1y_2''-y_1''y_2$, but at this point, I'm not too sure what to show.

Answer 1

you have $$y_1'' + py_1' + y_2 = 0\tag 1$$ $$y_2'' + py_2' + qy_2 = 0 \tag 2 $$ multiplying $(1)$ by $y_2,(2)$ by $y_1$ and subtracting gives you, $$y_1''y_2 - y_2y_1'' + p(y_1'y_2 - y_2'y_1) = 0 \to W' + pW = 0$$ the solution is $$W = Ce^{\int_0^x p\, dt} $$