Paper 2, Section I, A

Differential Equations | Part IA, 2021

Let y1y_{1} and y2y_{2} be two linearly independent solutions to the differential equation

d2y dx2+p(x)dy dx+q(x)y=0.\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}+p(x) \frac{\mathrm{d} y}{\mathrm{~d} x}+q(x) y=0 .

Show that the Wronskian W=y1y2y2y1W=y_{1} y_{2}^{\prime}-y_{2} y_{1}^{\prime} satisfies

dW dx+pW=0.\frac{\mathrm{d} W}{\mathrm{~d} x}+p W=0 .

Deduce that if y2(x0)=0y_{2}\left(x_{0}\right)=0 then

y2(x)=y1(x)x0xW(t)y1(t)2 dt.y_{2}(x)=y_{1}(x) \int_{x_{0}}^{x} \frac{W(t)}{y_{1}(t)^{2}} \mathrm{~d} t .

Given that y1(x)=x3y_{1}(x)=x^{3} satisfies the equation

x2d2y dx2xdy dx3y=0x^{2} \frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}-x \frac{\mathrm{d} y}{\mathrm{~d} x}-3 y=0

find the solution which satisfies y(1)=0y(1)=0 and y(1)=1y^{\prime}(1)=1.

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