Show that the substitution
$$u = \frac { \mathrm { d } y } { \mathrm {~d} x } + 2 y$$
transforms the differential equation
$$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 4 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 4 y = \mathrm { e } ^ { - 2 x }$$
into
$$\frac { \mathrm { d } u } { \mathrm {~d} x } + 2 u = \mathrm { e } ^ { - 2 x }$$
(4 marks)
By using an integrating factor, or otherwise, find the general solution of
$$\frac { \mathrm { d } u } { \mathrm {~d} x } + 2 u = \mathrm { e } ^ { - 2 x }$$
giving your answer in the form \(u = \mathrm { f } ( x )\).
Hence find the general solution of the differential equation
$$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 4 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 4 y = \mathrm { e } ^ { - 2 x }$$
giving your answer in the form \(y = \mathrm { g } ( x )\).