91Ó°ÊÓ

Find general solutions of the linear systems in Problems 1 through 20. If initial conditions are given, find the particular solution that satisfies them. In Problems I through 6, use a computer system or graphing calculator to construct a direction field and typical solution curves for the given system. \(x^{\prime}=-x+3 y, y^{\prime}=2 y\)

First calculate the operational determinant of the given system in order to determine how many arbitrary constants should appear in a general solution. Then attempt to solve the system explicitly so as to find such a general solution. \(\left(D^{2}+D\right) x+D^{2} y=2 e^{-t}\) \(\left(D^{2}-1\right) x+\left(D^{2}-D\right) y=0\)

A particle of mass \(m\) moves in the plane with coordinates \((x(t), y(t))\) under the influence of a force that is directeo toward the origin and has magnitude \(k /\left(x^{2}+y^{2}\right)-\) ar inverse-square central force field. Show that $$ m x^{\prime \prime}=-\frac{k x}{r^{3}} \quad \text { and } \quad m y^{\prime \prime}=-\frac{k y}{r^{3}}, $$ where \(r=\sqrt{x^{2}+y^{2}}\)