Convert the initial value problem $y''+\lambda x^2y=f(x)$ with $y'(0)=y(0)=0$ into a Volterra integral equation.
Here if we consider the two linearly independent solution of the above equation as $u(x)$ and $v(x)$, then we know that the reduced equation will be $$y(x)=\int_0^x \frac{u(t)v(x)-u(x)v(t)}{A}f(t)dt$$ where $A$ is given by $u(x)v'(x)-u'(x)v(x)=A$. But I could not find the solution of the above equation in terms of $u$ and $v$ explicitly. The method of inspection also did not help me in this regard. Can someone supply a method such that this type of equation can be solved easily? I know that this problem can be handled by Leibniz rule, but I want a help for this technique only. Thanks in advance.
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