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Expert-verified Found in: Page 180 ### Fundamentals Of Differential Equations And Boundary Value Problems

Book edition 9th
Author(s) R. Kent Nagle, Edward B. Saff, Arthur David Snider
Pages 616 pages
ISBN 9780321977069 # Determine the form of a particular solution for the differential equation. (Do not evaluate coefficients.) ${\mathbf{y}}{\mathbf{\text{'}}}{\mathbf{\text{'}}}{\mathbf{-}}{\mathbf{6}}{\mathbf{y}}{\mathbf{\text{'}}}{\mathbf{+}}{\mathbf{9}}{\mathbf{y}}{\mathbf{=}}{\mathbf{5}}{{\mathbf{t}}}^{6}{{\mathbf{e}}}^{3t}$

The particular solution is:

${\mathbf{y}}_{p}\left(\mathbf{x}\right)\mathbf{=}\left({\mathbf{A}}_{6}{\mathbf{t}}^{8}\mathbf{+}{\mathbf{A}}_{5}{\mathbf{t}}^{7}\mathbf{+}{\mathbf{A}}_{4}{\mathbf{t}}^{6}\mathbf{+}{\mathbf{A}}_{3}{\mathbf{t}}^{5}\mathbf{+}{\mathbf{A}}_{2}{\mathbf{t}}^{4}\mathbf{+}{\mathbf{A}}_{1}{\mathbf{t}}^{3}\mathbf{+}{\mathbf{A}}_{0}{\mathbf{t}}^{2}\right){\mathbf{e}}^{3t}$

See the step by step solution

## Step 1: Use the method of undetermined coefficients to find a particular solution of given differential equation.

The given differential equation is in the form of;

$\mathrm{ax}\text{'}\text{'}+\mathrm{bx}\text{'}+\mathrm{cx}={\mathrm{e}}^{\mathrm{rt}}$

To find a particular solution to the differential equation

$\mathrm{ay}\text{'}\text{'}\left(\mathrm{x}\right)+\mathrm{by}\text{'}\left(\mathrm{x}\right)+\mathrm{cy}\left(\mathrm{x}\right)={\mathrm{Ct}}^{\mathrm{m}}{\mathrm{e}}^{\mathrm{rt}}$

Where m is a nonnegative integer, use the form;

${\mathrm{y}}_{\mathrm{p}}\left(\mathrm{x}\right)={\mathrm{t}}^{\mathrm{s}}\left({\mathrm{A}}_{\mathrm{m}}{\mathrm{t}}^{\mathrm{m}}+...+{\mathrm{A}}_{1}\mathrm{t}+{\mathrm{A}}_{0}\right){\mathrm{e}}^{\mathrm{rt}}$

1. s = 0 if r is not a root of the associated auxiliary equation;
2. s = 1 if r is a simple root of the associated auxiliary equation;
3. s = 2 if r is a double root of the associated auxiliary equation.

## Step 2: Now, write the auxiliary equation of the above differential equation

The given differential equation is,

$\mathrm{y}\text{'}\text{'}-6\mathrm{y}\text{'}+9\mathrm{y}=5{\mathrm{t}}^{6}{\mathrm{e}}^{3\mathrm{t}}\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}......\left(1\right)$

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

$\mathrm{y}\text{'}\text{'}-6\mathrm{y}\text{'}+9\mathrm{y}=0$

The auxiliary equation for the above equation,

${\mathrm{r}}^{2}-6\mathrm{r}+9=0$

## Step 3: Now find the roots of auxiliary equation

Solve the auxiliary equation,

$\begin{array}{c}{\mathrm{r}}^{2}-6\mathrm{r}+9=0\\ {\left(\mathrm{r}-3\right)}^{2}=0\end{array}$

The roots of auxiliary equation are,

${\mathrm{r}}_{1}=3,\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}&\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}{\mathrm{r}}_{2}=3$

The complimentary solution of the given equation is,

${\mathrm{y}}_{\mathrm{c}}={\mathrm{c}}_{1}{\mathrm{e}}^{3\mathrm{t}}+{\mathrm{c}}_{2}{\mathrm{te}}^{3\mathrm{t}}$

## Step 4: Final conclusion

To find a particular solution to the differential equation;

$\mathrm{ay}\text{'}\text{'}\left(\mathrm{x}\right)+\mathrm{by}\text{'}\left(\mathrm{x}\right)+\mathrm{cy}\left(\mathrm{x}\right)={\mathrm{Ct}}^{\mathrm{m}}{\mathrm{e}}^{\mathrm{rt}}$

Compare with the given differential equation,

$\mathrm{y}\text{'}\text{'}-6\mathrm{y}\text{'}+9\mathrm{y}=5{\mathrm{t}}^{6}{\mathrm{e}}^{3\mathrm{t}}$

Condition satisfied,

M=6, s = 2 if r = 3 is a double root of the associated auxiliary equation.

Therefore, the particular solution of equation,

$\begin{array}{c}{\mathrm{y}}_{\mathrm{p}}\left(\mathrm{x}\right)={\mathrm{t}}^{2}\left({\mathrm{A}}_{6}{\mathrm{t}}^{6}+{\mathrm{A}}_{5}{\mathrm{t}}^{5}+{\mathrm{A}}_{4}{\mathrm{t}}^{4}+{\mathrm{A}}_{3}{\mathrm{t}}^{3}+{\mathrm{A}}_{2}{\mathrm{t}}^{2}+{\mathrm{A}}_{1}\mathrm{t}+{\mathrm{A}}_{0}\right){\mathrm{e}}^{3\mathrm{t}}\\ {\mathrm{y}}_{\mathrm{p}}\left(\mathrm{x}\right)=\left({\mathrm{A}}_{6}{\mathrm{t}}^{8}+{\mathrm{A}}_{5}{\mathrm{t}}^{7}+{\mathrm{A}}_{4}{\mathrm{t}}^{6}+{\mathrm{A}}_{3}{\mathrm{t}}^{5}+{\mathrm{A}}_{2}{\mathrm{t}}^{4}+{\mathrm{A}}_{1}{\mathrm{t}}^{3}+{\mathrm{A}}_{0}{\mathrm{t}}^{2}\right){\mathrm{e}}^{3\mathrm{t}}\end{array}$ ### Want to see more solutions like these? 