91Ó°ÊÓ

Find the general solution of each of the differential equations. $$ 2 y^{\prime \prime}+3 y^{\prime}-2 y=0. $$

Given that \(y=e^{2 x}\) is a solution of $$ (2 x+1) y^{\prime \prime}-4(x+1) y^{\prime}+4 y=0 $$ find a linearly independent solution by reducing the order. Write the general solution.

Given that \(y=x\) is a solution of $$ \left(x^{2}-2 x+2\right) y^{\prime \prime}-x^{2} y^{\prime}+x y=0 $$ find a linearly independent solution by reducing the order. Write the general solution.

Find the general solution of each of the differential equations. $$ 4 y^{\prime \prime}-4 y^{\prime}+y=0. $$

Find the general solution of each of the differential equations. $$ y^{\prime \prime}-4 y^{\prime}+4 y=0 $$

Given that \(y=e^{x}\) is a solution of $$ \left(x^{2}+x\right) y^{\prime \prime}-\left(x^{2}-2\right) y^{\prime}-(x+2) y=0 $$ find a linearly independent solution by reducing the order. Write the general

Find the general solution of each of the differential equations. $$ y^{\prime \prime}+6 y^{\prime}+11 y=0. $$

Find the general solution of each of the differential equations. $$ 6 y^{\prime \prime}+32 y^{\prime}+25 y=0. $$

Consider the nonhomogeneous differential equation $$ y^{\prime \prime}-3 y^{\prime}+2 y=4 x^{2} $$ (a) Show that \(e^{x}\) and \(e^{2 x}\) are linearly independent solutions of the corresponding homogeneous equation $$ y^{\prime \prime}-3 y^{\prime}+2 y=0 $$ (b) What is the complementary function of the given nonhomogeneous equation? (c) Show that \(2 x^{2}+6 x+7\) is a particular integral of the given equation. (d) What is the general solution of the given equation?

Given that a particular integral of $$ y^{\prime \prime}-5 y^{\prime}+6 y=1 \quad \text { is } \quad y=\frac{1}{6} $$ a particular integral of $$ y^{\prime \prime}-5 y^{\prime}+6 y=x \text { is } y=\frac{x}{6}+\frac{5}{36} \text { , } $$ and a particular integral of $$ y^{\prime \prime}-5 y^{\prime}+6 y=e^{x} \text { is } y=\frac{e^{x}}{2} $$ use Theorem \(4.10\) to find a particular integral of $$ y^{\prime \prime}-5 y^{\prime}+6 y=2-12 x+6 e^{x} \text { . } $$