Find a particular solution to the differential equation.

$\mathbf{2}\mathbf{z}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{z}\mathbf{=}\mathbf{9}{\mathbf{e}}^{2t}$

The given differential equation is,

$2\mathrm{z}\text{'}\text{'}+\mathrm{z}=9{\mathrm{e}}^{2\mathrm{t}}\dots \left(1\right)$

Write the homogeneous differential equation of the equation (1),

$2\mathrm{z}\text{'}\text{'}+\mathrm{z}=0$

The auxiliary equation for the above equation,

$2{\mathrm{m}}^2+1=0$

Solve the auxiliary equation,

$\begin{array}{c}2{\mathrm{m}}^2+1=0\\ {\mathrm{m}}^2=\frac{-1}{2}\\ \mathrm{m}=\pm \mathrm{i}\sqrt{\frac{1}{2}}\end{array}$

The roots of the auxiliary equation are,

${\mathrm{m}}_1=\mathrm{i}\sqrt{\frac{1}{2}},{\mathrm{m}}_2=-\mathrm{i}\sqrt{\frac{1}{2}}$

The complementary solution of the given equation is,

${\mathbf{Z}}_c\left(\mathbf{x}\right)\mathbf{=}{\mathbf{c}}_1\mathbf{cos}\left(\sqrt{\frac{1}{2}}\mathbf{t}\right)\mathbf{+}{\mathbf{c}}_2\mathbf{sin}\left(\sqrt{\frac{1}{2}}\mathbf{t}\right)$

According to the method of undetermined coefficients, assume, the particular solution of equation (1),

${\mathrm{z}}_{\mathrm{p}}\left(\mathrm{x}\right)={\mathrm{Ae}}^{2\mathrm{t}}......\left(2\right)$

Now find the derivative of the above equation,

$\begin{array}{c}{\mathrm{z}}_{\mathrm{p}}\text{'}\left(\mathrm{x}\right)=2{\mathrm{Ae}}^{2\mathrm{t}}\\ {\mathrm{z}}_{\mathrm{p}}\text{'}\text{'}\left(\mathrm{x}\right)=4{\mathrm{Ae}}^{2\mathrm{t}}\end{array}$

From the equation (1),

$\begin{array}{c}2{\mathrm{z}}_{\mathrm{p}}\text{'}\text{'}+{\mathrm{z}}_{\mathrm{p}}=9{\mathrm{e}}^{2\mathrm{t}}\\ 2\left(4{\mathrm{Ae}}^{2\mathrm{t}}\right)+{\mathrm{Ae}}^{2\mathrm{t}}=9{\mathrm{e}}^{2\mathrm{t}}\\ 9{\mathrm{Ae}}^{2\mathrm{t}}=9{\mathrm{e}}^{2\mathrm{t}}\\ \mathrm{A}=1\end{array}$

Substitute the value of A in the equation (2),

${\mathrm{z}}_{\mathrm{p}}\left(\mathrm{x}\right)={\mathrm{e}}^{2\mathrm{t}}$

Therefore, the particular solution of equation (1),

${\mathrm{z}}_{\mathrm{p}}\left(\mathrm{x}\right)={\mathrm{e}}^{2\mathrm{t}}$