Paper 2, Section II, 7 A7 \mathrm{~A}

Differential Equations | Part IA, 2011

(a) Define the Wronskian WW of two solutions y1(x)y_{1}(x) and y2(x)y_{2}(x) of the differential equation

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

and state a necessary and sufficient condition for y1(x)y_{1}(x) and y2(x)y_{2}(x) to be linearly independent. Show that W(x)W(x) satisfies the differential equation

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

(b) By evaluating the Wronskian, or otherwise, find functions p(x)p(x) and q(x)q(x) such that ()(*) has solutions y1(x)=1+cosxy_{1}(x)=1+\cos x and y2(x)=sinxy_{2}(x)=\sin x. What is the value of W(π)?W(\pi) ? Is there a unique solution to the differential equation for 0x<0 \leqslant x<\infty with initial conditions y(0)=0,y(0)=1y(0)=0, y^{\prime}(0)=1 ? Why or why not?

(c) Write down a third-order differential equation with constant coefficients, such that y1(x)=1+cosxy_{1}(x)=1+\cos x and y2(x)=sinxy_{2}(x)=\sin x are both solutions. Is the solution to this equation for 0x<0 \leqslant x<\infty with initial conditions y(0)=y(0)=0,y(0)=1y(0)=y^{\prime \prime}(0)=0, y^{\prime}(0)=1 unique? Why or why not?

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