Question 8 - A-Level Maths

Exam Board	CAIE
Module	Further Paper 2 (Further Paper 2)
Year	2022
Session	November
Topic	First order differential equations (integrating factor)

Use the substitution $u = 1 - ( \theta - 1 ) ^ { 2 }$ to find $$\int \frac { \theta - 1 } { \sqrt { 1 - ( \theta - 1 ) ^ { 2 } } } \mathrm {~d} \theta$$
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 $y = \mathrm { f } ( \theta )$.
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