Question

Solve the system of differential equations =xz+1, =-xy for x=0.3 using 4th order R-K method with y(0)=0, z(0)=1.

Answer 1

$$y'(x) = xz(x) + 1, \quad z'(x) = -xy(x), \quad y(0) = 0, \quad z(0) = 1$$

First-order system of linear differential equations

Solution

$$y(x) = \sqrt{\pi} C\left(\frac{x}{\sqrt{\pi}}\right) \cos\left(\frac{x^2}{2}\right) + \left(\sqrt{\pi} S\left(\frac{x}{\sqrt{\pi}}\right) + 1\right) \sin\left(\frac{x^2}{2}\right)$$ $$z(x) = -\sqrt{\pi} C\left(\frac{x}{\sqrt{\pi}}\right) \sin\left(\frac{x^2}{2}\right) + \sqrt{\pi} S\left(\frac{x}{\sqrt{\pi}}\right) \cos\left(\frac{x^2}{2}\right) + \cos\left(\frac{x^2}{2}\right)$$

where

$C(x)$ is the Fresnel C integral

$S(x)$ is the Fresnel S integral

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