Question

Oscillations and Nonlinear Equations. For the initial value problem $\mathbf{x}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{(}\mathbf{0}\mathbf{.}\mathbf{1}\mathbf{\left)}\mathbf{\right(}\mathbf{1}\mathbf{-}{\mathbf{x}}^{\mathbf{2}}\mathbf{)}\mathbf{x}\mathbf{\text{'}}\mathbf{+}\mathbf{x}\mathbf{=}\mathbf{0}\mathbf{;}\mathbf{x}\mathbf{(}\mathbf{0}\mathbf{)}\mathbf{=}{\mathbf{x}}_{\mathbf{o}}\mathbf{,}\mathbf{x}\mathbf{\text{'}}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{0}$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 ${\mathbf{x}}_{\mathbf{o}}\mathbf{=}\mathbf{2}\mathbf{.}\mathbf{1}\mathbf{,}$.

Short answer

The result can get by the Runge-Kutta method.

Step-by-step solution

Transform the equation

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

The system can be written as:

$$\begin{gathered} {\mathrm{x}}_1=\mathrm{x}\left(\mathrm{t}\right) \\ {\mathrm{x}}_2=\mathrm{x}\text{'}=\mathrm{x}{\text{'}}_1 \end{gathered}$$

The transform equation is:

role="math" localid="1664101777238" $$\begin{gathered} \mathbf{x}{\mathbf{\text{'}}}_{\mathbf{1}}\mathbf{=}{\mathbf{x}}_{\mathbf{2}} \\ \mathbf{x}{\mathbf{\text{'}}}_{\mathbf{2}}\mathbf{=}\mathbf{-}{\mathbf{x}}_{\mathbf{1}}\mathbf{-}\mathbf{0}\mathbf{.}\mathbf{1}\mathbf{(}\mathbf{1}\mathbf{-}{{\mathbf{x}}^{\mathbf{2}}}_{\mathbf{1}}\mathbf{)}{\mathbf{x}}_{\mathbf{2}} \end{gathered}$$

The initial conditions are,

$$\begin{gathered} {\mathrm{x}}_1\left(0\right)=\mathrm{x}\left(0\right)={\mathrm{x}}_{\mathrm{o}}=1,2,1 \\ {\mathrm{x}}_2\left(0\right)=\mathrm{x}\text{'}\left(0\right)=0 \end{gathered}$$

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.