# Paper 2, Section II, $7 \mathrm{~A}$

(a) Define the Wronskian $W$ of two solutions $y_{1}(x)$ and $y_{2}(x)$ of the differential equation

$y^{\prime \prime}+p(x) y^{\prime}+q(x) y=0$

and state a necessary and sufficient condition for $y_{1}(x)$ and $y_{2}(x)$ to be linearly independent. Show that $W(x)$ satisfies the differential equation

$W^{\prime}(x)=-p(x) W(x)$

(b) By evaluating the Wronskian, or otherwise, find functions $p(x)$ and $q(x)$ such that $(*)$ has solutions $y_{1}(x)=1+\cos x$ and $y_{2}(x)=\sin x$. What is the value of $W(\pi) ?$ Is there a unique solution to the differential equation for $0 \leqslant x<\infty$ with initial conditions $y(0)=0, y^{\prime}(0)=1$ ? Why or why not?

(c) Write down a third-order differential equation with constant coefficients, such that $y_{1}(x)=1+\cos x$ and $y_{2}(x)=\sin x$ are both solutions. Is the solution to this equation for $0 \leqslant x<\infty$ with initial conditions $y(0)=y^{\prime \prime}(0)=0, y^{\prime}(0)=1$ unique? Why or why not?