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Q4.3-2E

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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

# The auxiliary equation for the given differential equation has complex roots. Find a general solution ${\mathbf{y}}{\mathbf{\text{'}}}{\mathbf{\text{'}}}{\mathbf{+}}{\mathbf{y}}{\mathbf{=}}{\mathbf{0}}$.

The auxiliary equation for the given differential equation $\mathbf{y}\text{'}\text{'}\mathbf{+}\mathbf{y}\mathbf{=}\mathbf{0}$ has complex roots and its general solution is $\mathbf{y}\mathbf{=}{\mathbf{c}}_{1}\mathbf{cost}\mathbf{+}{\mathbf{c}}_{2}\mathbf{sint}$.

See the step by step solution

## Step 1: Complex conjugate roots.

If the auxiliary equation has complex conjugate roots $\mathrm{\alpha }±\mathrm{i\beta }$, then the general solution is given as:

$y\left(t\right)={c}_{1}{e}^{\alpha t}\mathrm{cos}\beta t+{c}_{2}{e}^{\alpha t}\mathrm{sin}\beta t$.

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

The auxiliary equation for $\mathrm{y}\text{'}\text{'}+\mathrm{y}=0$ is ${\mathrm{k}}^{2}+1=0$ . The solutions of the auxiliary equation are:

${\mathrm{k}}^{2}+1=0\phantom{\rule{0ex}{0ex}}{\mathrm{k}}^{2}=-1\phantom{\rule{0ex}{0ex}}\mathrm{k}=±\mathrm{i}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}$

Therefore, $\mathrm{\alpha }=0,\mathrm{\beta }=1$

$\mathrm{y}={\mathrm{c}}_{1}\mathrm{cost}+{\mathrm{c}}_{2}\mathrm{sint}$.