91Ó°ÊÓ

Find the general solution. $$y^{\prime \prime}+4 y^{\prime}+6 y=0$$

The method of undetermined coefficients can sometimes be used to solve first- order ordinary differential equations. Use the method to solve the equations in Exercises. $$y^{\prime}+y=\sin x$$

Find the general solution. $$16 y^{\prime \prime}-24 y^{\prime}+9 y=0$$

Solve the differential equations in Exercises subject to the given initial conditions. $$\frac{d^{2} y}{d x^{2}}+y=\sec ^{2} x, \quad-\frac{\pi}{2} < x < \frac{\pi}{2}, \quad y(0)=y^{\prime}(0)=1$$

Solve the differential equations in Exercises subject to the given initial conditions. $$\frac{d^{2} y}{d x^{2}}+y=e^{2 x} ; \quad y(0)=0, y^{\prime}(0)=\frac{2}{5}$$

Find the general solution. $$6 y^{\prime \prime}-5 y^{\prime}-6 y=0$$

In Exercises verify that the given function is a particular solution to the specified nonhomogeneous equation. Find the general solution, and evaluate its arbitrary constants to find the unique solution satisfying the equation and the given initial conditions. $$y^{\prime \prime}+y^{\prime}=x, \quad y_{\mathrm{p}}=\frac{x^{2}}{2}-x, \quad y(0)=0, y^{\prime}(0)=0$$

Find the general solution. $$9 y^{\prime \prime}+24 y^{\prime}+16 y=0$$

In Exercises verify that the given function is a particular solution to the specified nonhomogeneous equation. Find the general solution, and evaluate its arbitrary constants to find the unique solution satisfying the equation and the given initial conditions. $$y^{\prime \prime}+y=x, \quad y_{\mathrm{p}}=2 \sin x+x, \quad y(0)=0, y^{\prime}(0)=0$$

Find the general solution. $$4 y^{\prime \prime}+16 y^{\prime}+52 y=0$$