2.II.8B

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

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

and obtain a differential equation which exhibits its dependence on $x$. Explain the relevance of the Wronskian to the linear independence of $u_{1}$ and $u_{2}$.

Consider the equation

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

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

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