91Ó°ÊÓ

In Problems 1-36 find the general solution of the given differential equation. \(2 y^{\prime \prime}-3 y^{\prime}+4 y=0\)

In Problems 1-24 find a second solution of each differential equation. Use reduction of order or formula (5) as instructed. Assume an appropriate interval of validity. $$ \left(1-x^{2}\right) y^{\prime \prime}-2 x y^{\prime}=0 ; \quad y_{1}=1 $$

Solve, if possible, the given system of differential equations by either systematic elimination or determinants.\(D^{2} x-2\left(D^{2}+D\right) y=\sin t\) \(x+\quad D y=0\)

Solve the given differential equation. \(2 x^{2} y^{\prime \prime}+x y^{\prime}+y=0\)

Find the general solution of each differential equation. $$ 3 y^{\prime \prime \prime}+10 y^{\prime \prime}+15 y^{\prime}+4 y=0 $$

In Problems 1-36 find the general solution of the given differential equation. \(3 y^{\prime \prime}+2 y^{\prime}+y=0\)

In Problems 1-24 find a second solution of each differential equation. Use reduction of order or formula (5) as instructed. Assume an appropriate interval of validity. $$ x^{2} y^{\prime \prime}-x y^{\prime}+2 y=0 ; \quad y_{1}=x \sin (\ln x) $$

Find a linear differential operator that annihilates the given function. $$ 1+7 e^{2 x} $$

Solve the given differential equation by undetermined coefficients. $$ y^{\prime \prime}-2 y^{\prime}+5 y=e^{x} \cos 2 x $$

Solve, if possible, the given system of differential equations by either systematic elimination or determinants.\(D x=y\) \(D y=z\) \(D z=x\)