Q4.3-15E - Fundamentals Of Differential Equations And Boundary Value Problems

Question

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

Step-by-Step Solution

Answer

The general solution of the given equation $\mathrm{y}\text{'}\text{'}-3\mathrm{y}\text{'}-11\mathrm{y}=0$ is $\mathrm{y}\left(\mathrm{t}\right)={\mathrm{c}}_1{\mathrm{e}}^{{}^{\frac{(3+\sqrt{53})\mathrm{t}}{}}}+{\mathrm{c}}_2{\mathrm{e}}^{\frac{(3-\sqrt{53})\mathrm{t}}{}}$.

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:

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

Step 3: Final answer.

Therefore, the general solution is:

$\mathbf{y}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}{\mathbf{c}}_1{\mathbf{e}}^{{}^{\frac{(\mathbf{3}\mathbf{+}\sqrt{53})\mathbf{t}}{2}}}\mathbf{+}{\mathbf{c}}_2{\mathbf{e}}^{\frac{(\mathbf{3}\mathbf{-}\sqrt{53})\mathbf{t}}{2}}$