91Ó°ÊÓ

Replace each of the following differential equations by an equivalent system of first order equations: a. \(y^{\prime \prime}-x^{2} y^{\prime}-x y=0\); b. \(y^{\prime \prime \prime}=y^{\prime \prime}-x^{2}\left(y^{\prime}\right)^{2}\).

If a particle of mass \(m\) moves in the \(x y\) plane, its equations of motion are $$ m \frac{d^{2} x}{d t^{2}}=f(t, x, y) \quad \text { and } \quad m \frac{d^{2} y}{d t^{2}}=g(t, x, y) $$ where \(f\) and \(g\) represent the \(x\) and \(y\) components, respectively, of the force acting on the particle. Replace this system of two second order equations by an equivalent system of four first order equations of the form (1).

a. Show that $$ \left\\{\begin{array} { l } { x = e ^ { 4 t } } \\ { y = e ^ { 4 t } } \end{array} \quad \text { and } \quad \left\\{\begin{array}{l} x=e^{-2 t} \\ y=-e^{-2 t} \end{array}\right.\right. $$ are solutions of the homogeneous system $$ \left\\{\begin{array}{l} \frac{d x}{d t}=x+3 y \\ \frac{d y}{d t}=3 x+y \end{array}\right. $$ b Show in two ways that the given solutions of the system in (a) are linearly independent on every closed interval, and write the general solution of this system. c. Find the particular solution $$ \left\\{\begin{array}{l} x=x(t) \\ y=y(t) \end{array}\right. $$ of this system for which \(x(0)=5\) and \(y(0)=1\).

a. Show that $$ \left\\{\begin{array} { l } { x = 2 e ^ { 4 t } } \\ { y = 3 e ^ { 4 t } } \end{array} \text { and } \quad \left\\{\begin{array}{l} x=e^{-t} \\ y=-e^{-t} \end{array}\right.\right. $$ are solutions of the homogeneous system $$ \left\\{\begin{array}{l} \frac{d x}{d t}=x+2 y \\ \frac{d y}{d t}=3 x+2 y \end{array}\right. $$ b. Show in two ways that the given solutions of the system in (a) are linearly independent on every closed interval, and write the general solution of this system. c. Show that $$ \left\\{\begin{array}{l} x=3 t-2 \\ y=-2 t+3 \end{array}\right. $$ is a particular solution of the nonhomogeneous system $$ \left\\{\begin{array}{l} \frac{d x}{d t}=x+2 y+t-1 \\ \frac{d y}{d t}=3 x+2 y-5 t-2 \end{array}\right. $$ and write the general solution of this system.

a. Find the general solution of the system $$ \left\\{\begin{array}{l} \frac{d x}{d t}=x \\ \frac{d y}{d t}=y \end{array}\right. $$ b. Show that any second order equation obtained from the system in (a) is not equivalent to this system, in the sense that it has solutions that are not part of any solution of the system. Thus, although higher order equations are equivalent to systems, the reverse is not true, and systems are more general.