9231 P1 - Jun 2008 - Q11
6462
Show that, with a suitable value of the constant \(\alpha\), the substitution \(y=x^{\alpha} w\) reduces the differential equation
\(2 x^{2} \frac{\mathrm{~d}^{2} y}{\mathrm{~d} x^{2}}+\left(3 x^{2}+8 x\right) \frac{\mathrm{d} y}{\mathrm{~d} x}+\left(x^{2}+6 x+4\right) y=\mathrm{f}(x)\)
to
\(2 \frac{\mathrm{~d}^{2} w}{\mathrm{~d} x^{2}}+3 \frac{\mathrm{~d} w}{\mathrm{~d} x}+w=\mathrm{f}(x)\)
Find the general solution for \(y\) in the case where \(\mathrm{f}(x)=6 \sin 2 x+7 \cos 2 x\).
