Past exam of the mathematics course of the University of Cambridge / 2023 / ia / Paper 2 / 6A (Differential Equations)

The parameter sensitivity equation follows by differentiating with respect to $\mu$ and writing $u=y_{\mu }$:At $\mu =0$, $y_x=y$, $y(0)=1$, so $y(x,0)=e^x$. Thenand the integrating factor $e^{-x}$ yields

Equating powers in $y=y_0+\mu y_1+{\mu }^2{y}_2+\cdots$ givesConsequently