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Jun's question via email about Runge Kutta Scheme...

Prove It

Well-known member
MHB Math Helper
Jan 26, 2012
1,403
rks.png

You first have to write this DE as a system of first order equations.

Note, since $\displaystyle t$ does not appear in the original DE, that means that the system will be autonomous if kept in terms of $\displaystyle t$.

Let $\displaystyle y = u$ and $\displaystyle y' = v$, then

$\displaystyle \begin{align*}
y'' + 4\left( y' \right) ^2 - 7\,y &= 0.1 \\
y'' + 4\,v^2 - 7\,u &= 0.1 \\
y'' &= 7\,u - 4\,v^2 + 0.1
\end{align*}$

Thus the system is

$\displaystyle \begin{align*} u' &= v , \quad u\left( 0 \right) = 1 \\
v' &= 7\,u - 4\,v^2 + 0.1 , \quad v\left( 0 \right) = 0 \end{align*}$

So here the system has $\displaystyle f\left( u, v \right) = v$ and $\displaystyle g\left( u, v \right) = 7\,u - 4\,v^2 + 0.1$.

I have used my CAS to work through this question.

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Starting with $\displaystyle t = 0$, two steps of the scheme with stepsize $h = 0.1$ means that we are at $\displaystyle t = 0.2$, and since $\displaystyle y = u$ that means $\displaystyle y\left( 0.2 \right) = u_2 = 1.12317$.