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Expert-verified Found in: Page 231 ### 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 to the given differential equation.${\mathbf{u}}{\mathbf{\text{'}}}{\mathbf{\text{'}}}{\mathbf{+}}{\mathbf{11}}{\mathbf{u}}{\mathbf{=}}{\mathbf{0}}$

${\mathbf{u}}{\mathbf{=}}{{\mathbf{c}}}_{1}{\mathbf{cos}}\left(\sqrt{11}\mathbf{t}\right){\mathbf{+}}{{\mathbf{c}}}_{2}{\mathbf{sin}}\left(\sqrt{11}\mathbf{t}\right)$

See the step by step solution

## Step 1: Complex conjugate roots.

If the auxiliary equation has complex conjugate roots, then the general solution is given as: ${y}\left(t\right){=}{{c}}_{{1}}{{e}}^{\alpha t}{c}{o}{s}{\beta }{t}{+}{{c}}_{{2}}{{e}}^{\alpha t}{s}{i}{n}{\beta }{t}$

## 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,

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

The roots of the auxiliary equation are, ${\mathbf{m}}_{1}\mathbf{=}\mathbf{i}\sqrt{11}\mathbf{,}\text{\hspace{0.17em}\hspace{0.17em}}&\text{\hspace{0.17em}\hspace{0.17em}}{\mathbf{m}}_{2}\mathbf{=}\mathbf{-}\mathbf{i}\sqrt{11}.$

The general solution of the given equation is ${\mathbf{u}}{\mathbf{=}}{{\mathbf{c}}}_{1}{\mathbf{cos}}\left(\sqrt{11}\mathbf{t}\right){\mathbf{+}}{{\mathbf{c}}}_{2}{\mathbf{sin}}\left(\sqrt{11}\mathbf{t}\right){.}$ ### Want to see more solutions like these? 