Differential Equations | Part IA, 2005

Define the Wronskian W[u1,u2]W\left[u_{1}, u_{2}\right] for two solutions u1,u2u_{1}, u_{2} of the equation

d2udx2+p(x)dudx+q(x)u=0\frac{d^{2} u}{d x^{2}}+p(x) \frac{d u}{d x}+q(x) u=0

and obtain a differential equation which exhibits its dependence on xx. Explain the relevance of the Wronskian to the linear independence of u1u_{1} and u2u_{2}.

Consider the equation

x2d2ydx22y=0x^{2} \frac{d^{2} y}{d x^{2}}-2 y=0

and determine the dependence on xx of the Wronskian W[y1,y2]W\left[y_{1}, y_{2}\right] of two solutions y1y_{1} and y2y_{2}. Verify that y1(x)=x2y_{1}(x)=x^{2} is a solution of ()(*) and use the Wronskian to obtain a second linearly independent solution.

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