The variable $y$ depends on $x$, and the variables $x$ and $t$ are related by $x = \mathrm { e } ^ { t }$. Show that

$$x \frac { \mathrm {~d} y } { \mathrm {~d} x } = \frac { \mathrm { d } y } { \mathrm {~d} t } \quad \text { and } \quad x ^ { 2 } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = \frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } - \frac { \mathrm { d } y } { \mathrm {~d} t } .$$

(i) Given that $y$ satisfies the differential equation

$$4 x ^ { 2 } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 16 x \frac { \mathrm {~d} y } { \mathrm {~d} x } + 25 y = 50 ( \ln x ) - 1$$

find a differential equation involving only $t$ and $y$.\\
(ii) Show that the complementary function of the differential equation in $t$ and $y$ may be written in the form

$$R \mathrm { e } ^ { - \frac { 3 } { 2 } t } \sin ( 2 t + \phi )$$

where $R$ and $\phi$ are arbitrary constants.\\
(iii) Find a particular integral of the differential equation in $t$ and $y$.\\
(iv) Hence find the general solution of the differential equation in $x$ and $y$.

\hfill \mbox{\textit{CAIE FP1 2004 Q12 EITHER}}