Find the solution of the differential equation
$$\theta \frac { d y } { d \theta } - y = \theta ^ { 2 } \sin ^ { - 1 } ( \theta - 1 ) ,$$
where \(0 < \theta < 2\), given that \(y = 1\) when \(\theta = 1\). Give your answer in the form \(y = \mathrm { f } ( \theta )\).
If you use the following page to complete the answer to any question, the question number must be clearly shown.
8 (a) Use the substitution $u = 1 - ( \theta - 1 ) ^ { 2 }$ to find
$$\int \frac { \theta - 1 } { \sqrt { 1 - ( \theta - 1 ) ^ { 2 } } } \mathrm {~d} \theta$$
(b) Find the solution of the differential equation
$$\theta \frac { d y } { d \theta } - y = \theta ^ { 2 } \sin ^ { - 1 } ( \theta - 1 ) ,$$
where $0 < \theta < 2$, given that $y = 1$ when $\theta = 1$. Give your answer in the form $y = \mathrm { f } ( \theta )$.\\
If you use the following page to complete the answer to any question, the question number must be clearly shown.\\
\hfill \mbox{\textit{CAIE Further Paper 2 2022 Q8}}