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Q13E

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Found in: Page 332

### 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 for the differential equation with x as the independent variable: ${{y}}^{\left(4\right)}{+}{4}{y}{\text{'}}{\text{'}}{+}{4}{y}{=}{0}$

The general solution for the differential equation with x as the independent variable is

$y\left(x\right)={c}_{1}\mathrm{cos}\left(\sqrt{2}x\right)+{c}_{2}x\mathrm{cos}\left(\sqrt{2}x\right)+{c}_{3}\mathrm{sin}\left(\sqrt{2}x\right)+{c}_{4}x\mathrm{sin}\left(\sqrt{2}x\right)$

See the step by step solution

## Step 1: Auxiliary equation:

The auxiliary equation in this problem is ${r}^{4}+4{r}^{2}+4=0$ . This can be factored as ${\left({r}^{2}+2\right)}^{2}$ =0. Therefore this equation has roots $r=\sqrt{2}i,-\sqrt{2}i,\sqrt{2}i,-\sqrt{2}i$ , which we see are repeated and complex.

## Step 2: General solution:

The general solution to the given equation is given by

$y\left(x\right)={c}_{1}\mathrm{cos}\left(\sqrt{2}x\right)+{c}_{2}x\mathrm{cos}\left(\sqrt{2}x\right)+{c}_{3}\mathrm{sin}\left(\sqrt{2}x\right)+{c}_{4}x\mathrm{sin}\left(\sqrt{2}x\right)$

Hence the final solution is $y\left(x\right)={c}_{1}\mathrm{cos}\left(\sqrt{2}x\right)+{c}_{2}x\mathrm{cos}\left(\sqrt{2}x\right)+{c}_{3}\mathrm{sin}\left(\sqrt{2}x\right)+{c}_{4}x\mathrm{sin}\left(\sqrt{2}x\right)$

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