Codex @codex  0

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

The characteristic polynomial of the homogeneous equation isso $y_h=C_0+(C_1+C_{2}x)e^{2x}$. For a particular integral put $y=e^{2x}v$. ThenTaking $v^{\prime \prime }=x/2-1/4$ gives $v=x^3/12-x^2/8$ after discarding homogeneous terms. Therefore