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### 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 for the differential equation with x as the independent variable.${\mathbf{y}}{\mathbf{\text{'}}}{\mathbf{\text{'}}}{\mathbf{\text{'}}}{\mathbf{-}}{\mathbf{3}}{\mathbf{y}}{\mathbf{\text{'}}}{\mathbf{\text{'}}}{\mathbf{-}}{\mathbf{y}}{\mathbf{\text{'}}}{\mathbf{+}}{\mathbf{3}}{\mathbf{y}}{\mathbf{=}}{\mathbf{0}}$

Thus, the general solution to the given differential equation is; $\mathbf{y}\mathbf{=}{\mathbf{C}}_{1}{\mathbf{e}}^{3x}\mathbf{+}{\mathbf{C}}_{2}{\mathbf{e}}^{x}\mathbf{+}{\mathbf{C}}_{3}{\mathbf{e}}^{-x}$

See the step by step solution

## Step 1: Find the auxiliary equation

The given differential equation is;

$\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{-}\mathbf{3}\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{-}\mathbf{y}\mathbf{\text{'}}\mathbf{+}\mathbf{3}\mathbf{y}\mathbf{=}\mathbf{0}$

The auxiliary equation is,

${\mathbf{m}}^{3}\mathbf{-}\mathbf{3}{\mathbf{m}}^{2}\mathbf{-}\mathbf{m}\mathbf{+}\mathbf{3}\mathbf{=}\mathbf{0}$

Take a fraction of the above equation,

$\begin{array}{c}{\mathbf{m}}^{2}\left(\mathbf{m}\mathbf{-}\mathbf{3}\right)\mathbf{-}\mathbf{1}\left(\mathbf{m}\mathbf{-}\mathbf{3}\right)\mathbf{=}\mathbf{0}\\ \left(\mathbf{m}\mathbf{-}\mathbf{3}\right)\left({\mathbf{m}}^{2}\mathbf{-}\mathbf{1}\right)\mathbf{=}\mathbf{0}\\ {\mathbf{m}}_{1}\mathbf{=}\mathbf{3}\mathbf{,}\text{\hspace{0.17em}}{\mathbf{m}}_{2}\mathbf{=}\mathbf{1}\mathbf{,}\text{\hspace{0.17em}}{\mathbf{m}}_{3}\mathbf{=}\mathbf{-}\mathbf{1}\end{array}$

## Step 2: Write the general solution

The roots are real and distinct; therefore the general solution to the given differential equation is given as:

$\begin{array}{c}\mathbf{y}\mathbf{=}{\mathbf{C}}_{1}{\mathbf{e}}^{{\mathbf{m}}_{1}\mathbf{x}}\mathbf{+}{\mathbf{C}}_{2}{\mathbf{e}}^{{\mathbf{m}}_{2}\mathbf{x}}\mathbf{+}{\mathbf{C}}_{3}{\mathbf{e}}^{{\mathbf{m}}_{3}\mathbf{x}}\\ \mathbf{y}\mathbf{=}{\mathbf{C}}_{1}{\mathbf{e}}^{\left(\mathbf{3}\right)\mathbf{x}}\mathbf{+}{\mathbf{C}}_{2}{\mathbf{e}}^{\left(\mathbf{1}\right)\mathbf{x}}\mathbf{+}{\mathbf{C}}_{3}{\mathbf{e}}^{\left(\mathbf{-}\mathbf{1}\right)\mathbf{x}}\\ \mathbf{y}\mathbf{=}{\mathbf{C}}_{1}{\mathbf{e}}^{3x}\mathbf{+}{\mathbf{C}}_{2}{\mathbf{e}}^{x}\mathbf{+}{\mathbf{C}}_{3}{\mathbf{e}}^{-x}\end{array}$

Thus, the general solution to the given differential equation is;

$\mathbf{y}\mathbf{=}{\mathbf{C}}_{1}{\mathbf{e}}^{3x}\mathbf{+}{\mathbf{C}}_{2}{\mathbf{e}}^{x}\mathbf{+}{\mathbf{C}}_{3}{\mathbf{e}}^{-x}$