9 Given that $$x \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + ( 2 x + 2 ) \frac { \mathrm { d } y } { \mathrm {~d} x } + ( 2 - 3 x ) y = 10 \mathrm { e } ^ { 2 x }$$ and that \(v = x y\), show that $$\frac { \mathrm { d } ^ { 2 } v } { \mathrm {~d} x ^ { 2 } } + 2 \frac { \mathrm {~d} v } { \mathrm {~d} x } - 3 v = 10 \mathrm { e } ^ { 2 x }$$ Find the general solution for \(y\) in terms of \(x\).

9 Given that

$$x \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + ( 2 x + 2 ) \frac { \mathrm { d } y } { \mathrm {~d} x } + ( 2 - 3 x ) y = 10 \mathrm { e } ^ { 2 x }$$

and that $v = x y$, show that

$$\frac { \mathrm { d } ^ { 2 } v } { \mathrm {~d} x ^ { 2 } } + 2 \frac { \mathrm {~d} v } { \mathrm {~d} x } - 3 v = 10 \mathrm { e } ^ { 2 x }$$

Find the general solution for $y$ in terms of $x$.

\hfill \mbox{\textit{CAIE FP1 2014 Q9}}