Paper 2, Section II, 6A6 A

Differential Equations | Part IA, 2016

(a) The function y(x)y(x) satisfies

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

(i) Define the Wronskian W(x)W(x) of two linearly independent solutions y1(x)y_{1}(x) and y2(x)y_{2}(x). Derive a linear first-order differential equation satisfied by W(x)W(x).

(ii) Suppose that y1(x)y_{1}(x) is known. Use the Wronskian to write down a first-order differential equation for y2(x)y_{2}(x). Hence express y2(x)y_{2}(x) in terms of y1(x)y_{1}(x) and W(x)W(x).

(b) Verify that y1(x)=cos(xγ)y_{1}(x)=\cos \left(x^{\gamma}\right) is a solution of

axαy+bxα1y+y=0,a x^{\alpha} y^{\prime \prime}+b x^{\alpha-1} y^{\prime}+y=0,

where a,b,αa, b, \alpha and γ\gamma are constants, provided that these constants satisfy certain conditions which you should determine.

Use the method that you described in part (a) to find a solution which is linearly independent of y1(x)y_{1}(x).

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