Q10P - Fundamentals Of Differential Equations And Boundary Value Problems

Question

Find a general solution to the given differential equation.$\mathbf{u}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{11}\mathbf{u}\mathbf{=}\mathbf{0}$

Step-by-Step Solution

Answer

$\mathbf{u}\mathbf{=}{\mathbf{c}}_1\mathbf{cos}\left(\sqrt{11}\mathbf{t}\right)\mathbf{+}{\mathbf{c}}_2\mathbf{sin}\left(\sqrt{11}\mathbf{t}\right)$

Step 1: Complex conjugate roots.

If the auxiliary equation has complex conjugate roots, then the general solution is given as: $y\left(t\right)=c_1{e}^{\alpha t}cos\beta t+c_2{e}^{\alpha t}sin\beta t$

Step 2: Write the auxiliary equation of the given differential equation

The differential equation is,$\mathbf{u}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{11}\mathbf{u}\mathbf{=}\mathbf{0}$

 

The auxiliary equation for the above equation,${\mathbf{m}}^2\mathbf{+}\mathbf{11}\mathbf{=}\mathbf{0}$

Step 3: Find the roots of the auxiliary equation.

Solve the auxiliary equation,

 $\begin{array}{c}{\mathbf{m}}^2\mathbf{+}\mathbf{11}\mathbf{=}\mathbf{0}\\ {\mathbf{m}}^2\mathbf{=}\mathbf{-}\mathbf{11}\\ \mathbf{m}\mathbf{=}\mathbf{\pm }\mathbf{i}\sqrt{11}\end{array}$

 

The roots of the auxiliary equation are, ${\mathbf{m}}_1\mathbf{=}\mathbf{i}\sqrt{11}\mathbf{,}\&{\mathbf{m}}_2\mathbf{=}\mathbf{-}\mathbf{i}\sqrt{11}.$

The general solution of the given equation is $\mathbf{u}\mathbf{=}{\mathbf{c}}_1\mathbf{cos}\left(\sqrt{11}\mathbf{t}\right)\mathbf{+}{\mathbf{c}}_2\mathbf{sin}\left(\sqrt{11}\mathbf{t}\right).$