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Q5.3-22E

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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

# Oscillations and Nonlinear Equations. For the initial value problem $\mathbf{x}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{\left(}\mathbf{0}\mathbf{.}\mathbf{1}\mathbf{\right)}\mathbf{\left(}\mathbf{1}\mathbf{-}{{\mathbf{x}}}^{{\mathbf{2}}}\mathbf{\right)}\mathbf{x}\mathbf{\text{'}}\mathbf{+}\mathbf{x}\mathbf{=}\mathbf{0}\mathbf{;}\mathbf{x}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}{{\mathbf{x}}}_{{\mathbf{o}}}\mathbf{,}\mathbf{x}\mathbf{\text{'}}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{0}$ using the vectorized Runge–Kutta algorithm with h = 0.02 to illustrate that as t increases from 0 to 20, the solution x exhibits damped oscillations when ${{\mathbf{x}}}_{{\mathbf{o}}}{\mathbf{=}}{\mathbf{1}}$, whereas exhibits expanding oscillations when ${{\mathbf{x}}}_{{\mathbf{o}}}{\mathbf{=}}{\mathbf{2}}{\mathbf{.}}{\mathbf{1}}{\mathbf{,}}$.

The result can get by the Runge-Kutta method.

See the step by step solution

## Transform the equation

Here, the equation $\mathrm{x}\text{'}\text{'}+\left(0.1\right)\left(1-{\mathrm{x}}^{2}\right)\mathrm{x}\text{'}+\mathrm{x}=0.$

The system can be written as:

${\mathrm{x}}_{1}=\mathrm{x}\left(\mathrm{t}\right)\phantom{\rule{0ex}{0ex}}{\mathrm{x}}_{2}=\mathrm{x}\text{'}=\mathrm{x}{\text{'}}_{1}$

The transform equation is:

role="math" localid="1664101777238" ${\mathbf{x}}{{\mathbf{\text{'}}}}_{{\mathbf{1}}}{\mathbf{=}}{{\mathbf{x}}}_{{\mathbf{2}}}\phantom{\rule{0ex}{0ex}}{\mathbf{x}}{{\mathbf{\text{'}}}}_{{\mathbf{2}}}{\mathbf{=}}{\mathbf{-}}{{\mathbf{x}}}_{{\mathbf{1}}}{\mathbf{-}}{\mathbf{0}}\mathbf{.}\mathbf{1}\mathbf{\left(}\mathbf{1}\mathbf{-}{{\mathbf{x}}^{\mathbf{2}}}_{{\mathbf{1}}}{\mathbf{\right)}}{{\mathbf{x}}}_{{\mathbf{2}}}$

The initial conditions are,

${\mathrm{x}}_{1}\left(0\right)=\mathrm{x}\left(0\right)={\mathrm{x}}_{\mathrm{o}}=1,2,1\phantom{\rule{0ex}{0ex}}{\mathrm{x}}_{2}\left(0\right)=\mathrm{x}\text{'}\left(0\right)=0$

## Apply the Runge-Kutta method

Apply Matlab to find the results. And some results are;

 T For ${\mathrm{x}}_{0}=1$ For ${\mathrm{x}}_{\mathrm{o}}=2.1$ 0 1 2.1 0.02 0.9998 2.09957 0.04 0.9992 2.0983 0.06 0.9982 2.096 0.1 0.9950 2.0893 1 0.5441 1.0600 2 -0.36227 -0.9737

Applying the same procedure gets the result.

This is the required result.