91Ó°ÊÓ

Explain why the graph of the solution to the initial value problem \(y^{\prime}(t)=\frac{t^{2}}{1-t}, y(-1)=\ln 2\) cannot cross the line \(t=1\)

Find the general solution of the following equations. $$y^{\prime}(x)=-y+2$$

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

Verifying general solutions Verify that the given function is a solution of the differential equation that follows it. Assume \(C, C_{1}, C_{2}\) and \(C_{3}\) are arbitrary constants. $$y(t)=C e^{-5 t} ; y^{\prime}(t)+5 y(t)=0$$

Direction fields A differential equation and its direction field are shown in the following figures. Sketch a graph of the solution curve that passes through the given initial conditions. $$\begin{aligned}&y^{\prime}(t)=\frac{t^{2}}{y^{2}+1}, y(0)=-2\\\&\text { and } y(-2)=0\end{aligned}$$

Verifying general solutions Verify that the given function is a solution of the differential equation that follows it. Assume \(C, C_{1}, C_{2}\) and \(C_{3}\) are arbitrary constants. $$y(t)=C t^{3} ; t y^{\prime}(t)-3 y(t)=0$$

Describe the behavior of the two populations in a predator-prey model as functions of time.

Find the general solution of the following equations. $$y^{\prime}(x)=2 y+6$$

Find the general solution of the following equations. $$u^{\prime}(t)+12 u=15$$

Direction fields with technology Plot a direction field for the following differential equation with a graphing utility. Then find the solutions that are constant and determine which initial conditions \(y(0)=A\) lead to solutions that are increasing in time. $$y^{\prime}(t)=0.5(y+1)^{2}(t-1)^{2},|t| \leq 3 \text { and }|y| \leq 3$$