91Ó°ÊÓ

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. $$y^{\prime}(t)=(y-1) \sin \pi t, 0 \leq t \leq 2,0 \leq y \leq 2$$

Find the general solution of the following equations. $$v^{\prime}(y)-\frac{v}{2}=14$$

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. $$y(x)=C_{1} e^{-x}+C_{2} e^{x} ; y^{\prime \prime}(x)-y(x)=0$$

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. $$y^{\prime}(t)=t(y-1), 0 \leq t \leq 2,0 \leq y \leq 2$$

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

Solve the following initial value problems. $$y^{\prime}(t)=3 y-6, y(0)=9$$

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. $$u(t)=C e^{1 /\left(4 t^{4}\right)} ; u^{\prime}(t)+\frac{1}{t^{5}} u(t)=0$$

Solve the following initial value problems. $$y^{\prime}(x)=-y+2, y(0)=-2$$

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. $$y^{\prime}(t)=y-3, y(0)=1$$