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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.)${\mathbf{y}}{\mathbf{\text{'}}}{\mathbf{\text{'}}}{\mathbf{+}}{\mathbf{3}}{\mathbf{y}}{\mathbf{\text{'}}}{\mathbf{-}}{\mathbf{7}}{\mathbf{y}}{\mathbf{=}}{{\mathbf{t}}}^{4}{{\mathbf{e}}}^{t}$

The particular solution is ${\mathrm{y}}_{\mathrm{p}}\left(\mathrm{x}\right)=\left({\mathrm{A}}_{4}{\mathrm{t}}^{4}+{\mathrm{A}}_{3}{\mathrm{t}}^{3}+{\mathrm{A}}_{2}{\mathrm{t}}^{2}+{\mathrm{A}}_{1}\mathrm{t}+{\mathrm{A}}_{0}\right){\mathrm{e}}^{\mathrm{t}}$.

See the step by step solution

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

The given differential equation is in the form of;

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

According to the method of undetermined coefficients, 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 non-negative 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,

$\mathrm{y}\text{'}\text{'}+3\mathrm{y}\text{'}-7\mathrm{y}={\mathrm{t}}^{4}{\mathrm{e}}^{\mathrm{t}}\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}......\left(1\right)$

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

$\mathrm{y}\text{'}\text{'}+3\mathrm{y}\text{'}-7\mathrm{y}=0$

The auxiliary equation for the above equation,

${\mathrm{r}}^{2}+3\mathrm{r}-7=0$

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

Solve the auxiliary equation,

$\begin{array}{c}{\mathrm{r}}^{2}+3\mathrm{r}-7=0\\ \mathrm{r}=\frac{-3±\sqrt{{3}^{2}-4\left(1\right)\left(-7\right)}}{2\left(3\right)}\\ \mathrm{r}=\frac{-3±\sqrt{37}}{6}\end{array}$

The roots of the auxiliary equation are;

${\mathrm{r}}_{1}=\frac{-3+\sqrt{37}}{6},\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}{\mathrm{r}}_{2}=\frac{-3-\sqrt{37}}{6}$

The complementary solution of the given equation is;

${\mathrm{y}}_{\mathrm{c}}={\mathrm{c}}_{1}{\mathrm{e}}^{\frac{-3+\sqrt{37}}{6}\mathrm{t}}+{\mathrm{c}}_{2}{\mathrm{e}}^{\frac{-3-\sqrt{37}}{6}\mathrm{t}}$

## 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,

$\mathrm{y}\text{'}\text{'}+3\mathrm{y}\text{'}-7\mathrm{y}={\mathrm{t}}^{4}{\mathrm{e}}^{\mathrm{t}}$

Condition satisfied,

M=4, s = 0 if is not a root of the associated auxiliary equation;

Therefore, the particular solution of the equation,

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