91Ó°ÊÓ

For the systems of differential equations in Exercises \(7-12\), use Euler's method with \(\Delta t=2\) to a) Plot the graphs of \(x\) and \(y\) for \(0 \leq t \leq 500\). b) Plot the trajectory of \(x\) and \(y\). \(x^{\prime}=x(0.1-0.01 x-0.004 y)\) \(y^{\prime}=y(0.05-0.001 x-0.0016 y), x(0)=20\) \(y(0)=4\)

Find the general solution of the system of equations. \(x^{\prime}=2 x-y, y^{\prime}=4 x-2 y\)

Solve. \(2 y^{\prime \prime}-y^{\prime}-y=0\)

Find the equilibrium points and assess the stability of each. \(x^{\prime}=x-e^{y}, y^{\prime}=x+e^{2 y}-2\)

Verify by substitution that the given functions solve the system of differential equations. \(\left[\begin{array}{l}x \\\ y\end{array}\right]^{\prime}=\left[\begin{array}{rr}4 & 1 \\ -2 & 1\end{array}\right]\left[\begin{array}{l}x \\ y\end{array}\right]\) \(x=2 e^{3 t}-e^{2 t}, y=-2 e^{3 t}+2 e^{2 t}\)

For the systems of differential equations in Exercises \(7-12\), use Euler's method with \(\Delta t=2\) to a) Plot the graphs of \(x\) and \(y\) for \(0 \leq t \leq 500\). b) Plot the trajectory of \(x\) and \(y\). \(x^{\prime}=x(0.04-0.001 x-0.0022 y)\) \(y^{\prime}=y(0.02-0.0012 x-0.0004 y), x(0)=5\) \(y(0)=7\)

Solve. \(y^{\prime \prime}-9 y=0\)

Find the general solution of the system of equations. \(x^{\prime}=2 y, y^{\prime}=-18 x\)

Solve. \(y^{\prime \prime}-y^{\prime}-2 y=x^{3}-1\)

Find the equilibrium points and assess the stability of each. \(x^{\prime}=x-e^{y}, y^{\prime}=2 \ln x+y-6\)