Show that if \(y\) depends on \(x\) and \(x=\mathrm{e}^{u}\) then
\(x^{2} \frac{\mathrm{~d}^{2} y}{\mathrm{~d} x^{2}}=\frac{\mathrm{d}^{2} y}{\mathrm{~d} u^{2}}-\frac{\mathrm{d} y}{\mathrm{~d} u} .\)

Given that \(y\) satisfies the differential equation
\(x^{2} \frac{\mathrm{~d}^{2} y}{\mathrm{~d} x^{2}}+5 x \frac{\mathrm{~d} y}{\mathrm{~d} x}+3 y=30 x^{2}\)
use the substitution \(x=\mathrm{e}^{u}\) to show that
\(\frac{\mathrm{d}^{2} y}{\mathrm{~d} u^{2}}+4 \frac{\mathrm{~d} y}{\mathrm{~d} u}+3 y=30 \mathrm{e}^{2 u}\)

Hence find the general solution for \(y\) in terms of \(x\).