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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 # The auxiliary equation for the given differential equation has complex roots. Find a general solution. ${\mathbf{4}}{\mathbf{y}}{\mathbf{\text{'}}}{\mathbf{\text{'}}}{\mathbf{+}}{\mathbf{4}}{\mathbf{y}}{\mathbf{\text{'}}}{\mathbf{+}}{\mathbf{6}}{\mathbf{y}}{\mathbf{=}}{\mathbf{0}}$

The auxiliary equation for the given differential equation $4\mathrm{y}\text{'}\text{'}+4\mathrm{y}\text{'}+6\mathrm{y}=0$ has complex roots and its general solution is $\mathrm{y}\left(\mathrm{t}\right)={\mathrm{e}}^{-\frac{1}{2}\mathrm{t}}\left({\mathrm{c}}_{1}\mathrm{cos}\left(\frac{\sqrt{5}\mathrm{t}}{2}\right)\mathrm{t}+{\mathrm{c}}_{2}\mathrm{sin}\left(\frac{\sqrt{5}\mathrm{t}}{2}\right)\right)$ .

See the step by step solution

## Step 1: Complex conjugate roots.

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

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

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

The given differential equation is $4\mathrm{y}\text{'}\text{'}+4\mathrm{y}\text{'}+6\mathrm{y}=0.$

Then the auxiliary equation is $4{\mathrm{r}}^{2}+4\mathrm{r}+6=0.$

The roots of the auxiliary equation are:

role="math" localid="1654064879531" $\mathrm{r}=\frac{-4±\sqrt{{4}^{2}-4×4×6}}{2×4}\phantom{\rule{0ex}{0ex}}\mathrm{r}=\frac{-4±\sqrt{16-96}}{}\phantom{\rule{0ex}{0ex}}\mathrm{r}=\frac{-4±4\mathrm{i}\sqrt{5}}{}\phantom{\rule{0ex}{0ex}}\mathrm{r}=-\frac{1}{2}±\frac{\sqrt{5}\mathrm{i}}{2}$

Therefore, the general solution is:

$\mathrm{y}\left(\mathrm{t}\right)={\mathrm{e}}^{-\frac{1}{2}×\mathrm{t}}\left({\mathrm{c}}_{1}\mathrm{cos}\left(\frac{\sqrt{5}\mathrm{t}}{2}\mathrm{t}\right)+{\mathrm{c}}_{2}\mathrm{sin}\left(\frac{\sqrt{5}\mathrm{t}}{2}\right)\right)\phantom{\rule{0ex}{0ex}}\mathrm{y}\left(\mathrm{t}\right)={\mathrm{e}}^{-\frac{1}{2}\mathrm{t}}\left({\mathrm{c}}_{1}\mathrm{cos}\left(\frac{\sqrt{5}\mathrm{t}}{2}\mathrm{t}\right)+{\mathrm{c}}_{2}\mathrm{sin}\left(\frac{\sqrt{5}\mathrm{t}}{2}\right)\right)$ ### Want to see more solutions like these? 