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#### CaptainBlack

##### Well-known member

- Jan 26, 2012

- 890

**Part 2 of Pharaoh's Taylor series and modified Euler question from Yahoo Answers**

consider van der pol's equation

y" - 0.2(1-y^2)y' + y = 0 y(0)=0.1 y'(0)=0.1

2)

Write down the above problem as a system of first order differential equations.

Calculate the numerical solution at x = 0.2 using the Modified Euler method. Take the

step-length h = 0.1 and work to 6 decimal places accuracy. Compare with your solution

in part (i) and comment on your answers.

There is a standard method of converting higher order ODEs into first order systems, in this case it is to introduce the state vector:

\(Y(t)=\left[ \begin{array}{c} y(t) \\ y'(t) \end{array} \right] \)

Then the ODE becomes:

\( Y'(t)= F(t,Y)=\left[ \begin{array}{c} y'(t) \\ y''(t) \end{array} \right]= \left[ \begin{array}{c} y'(t) \\ 0.2 \left(1-(y(t))^2 \right) y'(t)-y(t) \end{array} \right] \)

Now you can use the standard form of the modified Euler method on this vector first order ODE.

CB