91Ó°ÊÓ

Use the window \([-2,2] \times[-2,2]\) to sketch a direction field for the following equations. Then sketch the solution curve that corresponds to the given initial condition. \(A\) detailed direction field is not needed. $$y^{\prime}(t)=y(2-y), y(0)=1$$

Verify that the given function \(y\) is a solution of the initial value problem that follows it. $$y(x)=\frac{1}{4}\left(e^{2 x}-e^{-2 x}\right) ; y^{\prime \prime}(x)-4 y(x)=0, y(0)=0, y^{\prime}(0)=1$$

Solve the following initial value problems. $$u^{\prime}(x)=2 u+6, u(1)=6$$

Find the general solution of the following equations. Express the solution explicitly as a function of the independent variable. $$y^{\prime}(t) e^{t / 2}=y^{2}+4$$

Make a sketch of the population function (as a function of time) that results from the following growth rate functions. Assume the population at time \(t=0\) begins at some positive value.

Find the general solution of each differential equation. Use \(C, C_{1}, C_{2}, \ldots .\) to denote arbitrary constants. $$y^{\prime}(t)=3+e^{-2 t}$$

Solve the following initial value problems. $$y^{\prime}(t)-3 y=12, y(1)=4$$

Use the window \([-2,2] \times[-2,2]\) to sketch a direction field for the following equations. Then sketch the solution curve that corresponds to the given initial condition. \(A\) detailed direction field is not needed. $$y^{\prime}(x)=\sin x, y(-2)=2$$

Write a logistic equation with the following parameter values. Then solve the initial value problem and graph the solution. Let \(r\) be the natural growth rate, \(K\) the carrying capacity, and \(P_{0}\) the initial population. $$r=0.2, K=300, P_{0}=50$$

Solve the following initial value problems. $$z^{\prime}(t)+\frac{z}{2}=6, z(-1)=0$$