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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 # Find a general solution ${\mathbf{y}}{\text{'}}{\text{'}}{\mathbf{-}}{\mathbf{3}}{\mathbf{y}}{\text{'}}{\mathbf{-}}{\mathbf{11}}{\mathbf{y}}{\mathbf{=}}{\mathbf{0}}$

The general solution of the given equation $\mathrm{y}\text{'}\text{'}-3\mathrm{y}\text{'}-11\mathrm{y}=0$ is $\mathrm{y}\left(\mathrm{t}\right)={\mathrm{c}}_{1}{\mathrm{e}}^{{}^{\frac{\left(3+\sqrt{53}\right)\mathrm{t}}{}}}+{\mathrm{c}}_{2}{\mathrm{e}}^{\frac{\left(3-\sqrt{53}\right)\mathrm{t}}{}}$.

See the step by step solution

## Step 1: Given information.

The differential equation is $\mathrm{y}\text{'}\text{'}-3\mathrm{y}\text{'}-11\mathrm{y}=0.$

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

Then the auxiliary equation is ${\mathrm{r}}^{2}-3\mathrm{r}-11=0.$

The roots of the auxiliary equation are:

$\begin{array}{c}\mathrm{r}=\frac{3±\sqrt{{3}^{2}-4×-11×1}}{2×1}\\ \mathrm{r}=\frac{3±\sqrt{9+44}}{2}\\ \mathrm{r}=\frac{3±\sqrt{53}}{2}\end{array}$

Therefore, the general solution is:

$\mathbf{y}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}{\mathbf{c}}_{1}{\mathbf{e}}^{{}^{\frac{\left(\mathbf{3}\mathbf{+}\sqrt{53}\right)\mathbf{t}}{2}}}\mathbf{+}{\mathbf{c}}_{2}{\mathbf{e}}^{\frac{\left(\mathbf{3}\mathbf{-}\sqrt{53}\right)\mathbf{t}}{2}}$ ### Want to see more solutions like these? 