91Ó°ÊÓ

Use the annihilator method to solve the given differential equation. $$y^{\prime \prime}+4 y=7 e^{x}.$$

Use the annihilator method to solve the given differential equation. $$y^{\prime \prime}+4 y=8 \cos 2 x.$$

Prove that the linear differential operator of order \(n\) $$ L=D^{n}+a_{1} D^{n-1}+\cdots+a_{n-1} D+a_{n} $$ is a linear transformation from \(C^{n}(I)\) to \(C^{0}(I)\)

Use the annihilator method to solve the given differential equation. $$y^{\prime \prime}-y=3 e^{2 x}+\sin x.$$

Derive an appropriate trial solution for the differential equation $$ P(D) y=c x^{k} e^{a x} \cos b x $$.

Use the variation-of-parameters method to solve the given differential equation. $$y^{\prime \prime}-2 y^{\prime}+y=e^{x} \ln x, \quad x > 0.$$

Find the general solution to the given differential equation on the interval \((0, \infty).\) $$x^{2} y^{\prime \prime}+9 x y^{\prime}+15 y=0.$$