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Q4.3-9E

Expert-verified
Found in: Page 172

### 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

# Find a general solution. ${\mathbf{y}}{\mathbf{\text{'}}}{\mathbf{\text{'}}}{\mathbf{-}}{\mathbf{8}}{\mathbf{y}}{\mathbf{\text{'}}}{\mathbf{+}}{\mathbf{7}}{\mathbf{y}}{\mathbf{=}}{\mathbf{0}}$

The general solution of the given equation $\mathrm{y}\text{'}\text{'}-8\mathrm{y}\text{'}+7\mathrm{y}=0$ is $\mathrm{y}\left(\mathrm{t}\right)={\mathrm{c}}_{1}{\mathrm{e}}^{\mathrm{t}}+{\mathrm{c}}_{2}{\mathrm{e}}^{7\mathrm{t}}$.

See the step by step solution

## Step 1: Differentiate the value of y.

Given differential equation is $\mathrm{y}\text{'}\text{'}-8\mathrm{y}\text{'}+7\mathrm{y}=0$

Let role="math" localid="1654066682579" $\mathrm{y}={\mathrm{e}}^{\mathrm{rt}}$

Therefore,

$\mathrm{y}\text{'}\left(\mathrm{t}\right)=\mathrm{r}{\mathrm{e}}^{\mathrm{rt}}\phantom{\rule{0ex}{0ex}}\mathrm{y}\text{'}\text{'}\left(\mathrm{t}\right)={\mathrm{r}}^{2}{\mathrm{e}}^{\mathrm{rt}}$

## Step 2: Finding the roots of the auxiliary equation.

Then the auxiliary equation is ${\mathrm{r}}^{2}-8\mathrm{r}+7=0.$

Now find the roots of the auxiliary equation.

role="math" localid="1654066897417" $\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}{\mathbf{r}}^{2}\mathbf{-}\mathbf{8}\mathbf{r}\mathbf{+}\mathbf{7}\mathbf{=}\mathbf{0}\phantom{\rule{0ex}{0ex}}{\mathbf{r}}^{2}\mathbf{-}\mathbf{r}\mathbf{-}\mathbf{7}\mathbf{r}\mathbf{+}\mathbf{7}\mathbf{=}\mathbf{0}\phantom{\rule{0ex}{0ex}}\mathbf{r}\mathbf{\left(}\mathbf{r}\mathbf{-}\mathbf{1}\mathbf{\right)}\mathbf{-}\mathbf{7}\mathbf{\left(}\mathbf{r}\mathbf{-}\mathbf{7}\mathbf{\right)}\mathbf{=}\mathbf{0}\phantom{\rule{0ex}{0ex}}\mathbf{\left(}\mathbf{r}\mathbf{-}\mathbf{1}\mathbf{\right)}\mathbf{\left(}\mathbf{r}\mathbf{-}\mathbf{7}\mathbf{\right)}\mathbf{=}\mathbf{0}$

$\mathbf{r}\mathbf{=}\mathbf{1}\mathbf{,}\mathbf{r}\mathbf{=}\mathbf{7}$

Therefore, the general solution is $\mathrm{y}\left(\mathrm{t}\right)={\mathrm{c}}_{1}{\mathrm{e}}^{\mathrm{t}}+{\mathrm{c}}_{2}{\mathrm{e}}^{7\mathrm{t}}$