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Q4.3-11E

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 ${\mathbf{z}}{\mathbf{"}}{\mathbf{+}}{\mathbf{10}}{\mathbf{z}}{\mathbf{\text{'}}}{\mathbf{+}}{\mathbf{25}}{\mathbf{z}}{\mathbf{=}}{\mathbf{0}}$

The general solution of the given equation $\mathrm{z}"+10\mathrm{z}\text{'}+25\mathrm{z}=0$ is $\mathrm{y}\left(\mathrm{t}\right)=\left({\mathrm{c}}_{1}+{\mathrm{c}}_{2}\mathrm{t}\right){\mathrm{e}}^{-5\mathrm{t}}$.

See the step by step solution

## Step 1: Given information.

Given differential equation is $\mathrm{z}"+10\mathrm{z}\text{'}+25\mathrm{z}=0$.

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

Then the auxiliary equation is ${\mathrm{r}}^{2}+10\mathrm{r}+25=0$

Solve the auxiliary equation to obtain the roots.

${\left(\mathrm{r}+5\right)}^{2}=0\phantom{\rule{0ex}{0ex}}\mathrm{r}+5=0\phantom{\rule{0ex}{0ex}}\mathrm{r}=-5$

## Step 3: Final answer.

Therefore, the general solution is $\mathrm{y}\left(\mathrm{t}\right)=\left({\mathrm{c}}_{1}+{\mathrm{c}}_{2}\mathrm{t}\right){\mathrm{e}}^{-5\mathrm{t}}$