5 Convergent infinite series \(C\) and \(S\) are defined by $$\begin{gathered} C = 1 + \frac { 1 } { 2 } \cos \theta + \frac { 1 } { 4 } \cos 2 \theta + \frac { 1 } { 8 } \cos 3 \theta + \ldots
S = \quad \frac { 1 } { 2 } \sin \theta + \frac { 1 } { 4 } \sin 2 \theta + \frac { 1 } { 8 } \sin 3 \theta + \ldots \end{gathered}$$

1. Show that \(C + \mathrm { i } S = \frac { 2 } { 2 - \mathrm { e } ^ { \mathrm { i } \theta } }\).
2. Hence show that \(C = \frac { 4 - 2 \cos \theta } { 5 - 4 \cos \theta }\), and find a similar expression for \(S\).
3. Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 2 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 17 y = 17 x + 36$$
4. Show that, when \(x\) is large and positive, the solution approximates to a linear function, and state its equation.