91Ó°ÊÓ

Find the general solution of each of the differential equations in exercise. $$ \frac{d^{3} y}{d x^{3}}-6 \frac{d^{2} y}{d x^{2}}+11 \frac{d y}{d x}-6 y=x e^{x}-4 e^{2 x}+6 e^{4 x} $$

Find the general solution of $$ x^{2} \frac{d^{2} y}{d x^{2}}-6 x \frac{d y}{d x}+10 y=3 x^{4}+6 x^{3} $$ given that \(y=x^{2}\) and \(y=x^{5}\) are linearly independent solutions of the corresponding homogeneous equation.

Find the general solution of $$ (x+1)^{2} \frac{d^{2} y}{d x^{2}}-2(x+1) \frac{d y}{d x}+2 y=1 $$ given that \(y=x+1\) and \(y=(x+1)^{2}\) are linearly independent solutions of the corresponding homogeneous equation.

Find the general solution of each of the differential equations in exercise. $$ \frac{d^{4} y}{d x^{4}}-\frac{d^{3} y}{d x^{3}}-3 \frac{d^{2} y}{d x^{2}}+\frac{d y}{d x}+2 y=0. $$

Find the general solution of $$ \left(x^{2}+2 x\right) \frac{d^{2} y}{d x^{2}}-2(x+1) \frac{d y}{d x}+2 y=(x+2)^{2} $$ given that \(y=x+1\) and \(y=x^{2}\) are linearly independent solutions of the corresponding homogeneous equation.

Find the general solution of each of the differential equations in exercise. $$ \frac{d^{2} y}{d x^{2}}+y=x \sin x $$.

Solve the initial-value problem in each of exercise. In each case assume \(x>0\). $$ x^{2} \frac{d^{2} y}{d x^{2}}-4 x \frac{d y}{d x}+6 y=0, \quad y(2)=0, \quad y^{\prime}(2)=4 $$.

Find the general solution of each of the differential equations in exercise. $$ \frac{d^{4} y}{d x^{4}}+6 \frac{d^{3} y}{d x^{3}}+15 \frac{d^{2} y}{d x^{2}}+20 \frac{d y}{d x}+12 y=0. $$

Find the general solution of $$ x^{2} \frac{d^{2} y}{d x^{2}}-x(x+2) \frac{d y}{d x}+(x+2) y=x^{3} $$ given that \(y=x\) and \(y=x e^{\star}\) are linearly independent solutions of the corresponding homogeneous equation.

Find the general solution of each of the differential equations in exercise. $$ \frac{d^{4} y}{d x^{4}}+y=0. $$