- (a) Show that the transformation \(x = \mathrm { e } ^ { u }\) transforms the differential equation
$$x ^ { 2 } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 7 x \frac { \mathrm {~d} y } { \mathrm {~d} x } + 16 y = 2 \ln x , \quad x > 0$$
into the differential equation
$$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} u ^ { 2 } } - 8 \frac { \mathrm {~d} y } { \mathrm {~d} u } + 16 y = 2 u$$
(b) Find the general solution of the differential equation (II), expressing \(y\) as a function of \(u\).
(c) Hence obtain the general solution of the differential equation (I).