91Ó°ÊÓ

Verify that the following are solutions of the differential equations given: (a) \(\phi(x)=e^{-\sin x}\), for \(y^{\prime}+(\cos x) y=0\) (b) \(\phi(x)=\sin x-1\), for \(y^{\prime}+(\cos x) y=\sin x \cos x\) (c) \(\phi(x)=1\), for \(y^{\prime \prime}-y^{\prime}=0\) (d) \(\phi(x)=e^{x}\), for \(y^{\prime \prime}-y^{\prime}=0\) (e) \(\phi(x)=c_{1}+c_{2} e^{x}\), for \(y^{\prime \prime}-y^{\prime}=0,\left(c_{1}, c_{2}\right.\) any constants) (f) \(\phi(x)=\sin 2 x\), for \(y^{\prime \prime}+4 y=0\) (g) \(\phi(x)=e^{2 i x}\), for \(y^{\prime \prime}+4 y=0\) (h) \(\phi(x)=c_{1} \cos k x+c_{2} \sin k x\), for \(y^{\prime \prime}+k^{2} y=0,\left(k\right.\) a positive constant, and \(c_{1}, c_{2}\) any constants)

5\. The equation $$ y^{\prime}+\alpha(x) y=\beta(x) y^{k}, \quad(k \text { constant) } $$ is called Bernoulli's equation. (a) Show that the formal substitution \(z=y^{1-k}\) transforms this into the linear equation $$ z^{\prime}+(1-k) \alpha(x) z=(1-k) \beta(x) $$ (b) Find all solutions of \(y^{\prime}-2 x y=x y^{2}\).