Chapter 1.11, Problem 19P

Consider the general third-order differential equation of the form

$y^{‴}=F\left(x,y^{″}\right)$

$\left(\mathrm{1.11.27}\right)$

(a) Show that Equation $\left(\mathrm{1.11.27}\right)$ can be replaced by the equivalent first-order system $\frac{d{u}_1}{dx}=u_2$, $\frac{d{u}_2}{dx}=u_3$, $\frac{d{u}_3}{dx}=F\left(x,u_3\right)$ where the variables $u_1$, $u_2$, $u_3$ are defined by $u_1=y$, $u_2=y^{\prime }$, $u_3=y^{″}$.

(b) Solve $y^{‴}=x^{-1}\left(y^{″}-1\right)$.