91Ó°ÊÓ

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

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

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. $$y^{\prime \prime}+y^{\prime}=x, \quad y_{\mathrm{p}}=\frac{x^{2}}{2}-x, \quad y(0)=0, y^{\prime}(0)=0$$

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. $$4 y^{\prime \prime}+16 y^{\prime}+52 y=0$$

Find the general solution. $$6 y^{\prime \prime}-5 y^{\prime}-4 y=0$$

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. $$y^{\prime \prime}-y^{\prime}-2 y=1-2 x, \quad y_{\mathrm{p}}=x-1, \quad y(0)=0, y^{\prime}(0)=1$$

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

Solve the initial value problem. $$y^{\prime \prime}+2 y^{\prime}+y=0, \quad y(0)=1, y^{\prime}(0)=1$$

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. $$y^{\prime \prime}-2 y^{\prime}+y=2 e^{x}, \quad y_{\mathrm{p}}=x^{2} e^{x}, \quad y(0)=1, y^{\prime}(0)=0$$