#### Solving Ordinary Differential Equations: Screencasts

Demonstrates how the Euler method is used to solve ordinary differential equations (ODEs).

We suggest you list the important points in this screencast as a way to increase retention.

Reviews how the Runge-Kutta method is used to solve ordinary differential equations.

We suggest you list the important points in this screencast as a way to increase retention.

##### Important Equations:

Euler method for the ODE $$\frac{dx}{dy}\,=\,f(x,y)$$ with the initial condition at $$x\,=\,x_0,y\,=y_0$$

The Euler method uses a step-by-step method where the $$n^{th}$$ step is: $y_{n+1}\,=\,y_n+(x_{n+1}-x_n)f(x_n,y_n)\,=\,y_n+hf(x_n,y_n)$

where $$h$$ is the step size, which is constant.

Runge-Kutta method for the ODE $$\frac{dx}{dy}\,=\,f(x,y)$$ with the initial condition at $$x\,=\,x_0,y\,=y_0$$ $y_{n+1}\,=\,y_n+hfT_4(x_n,y_n,h)\;\;\;\;x_{n+1}\,=\,x_n+h$

where $$T_4\,=\,\frac{1}{6}(k_1+2k_2+2k_3+k_4)$$ $k_1\,=\,f(x,y)$ $k_2\,=\,f(x+\frac{h}{2},y+\frac{h}{2}k_1)$ $k_3\,=\,f(x+\frac{h}{2},y+\frac{h}{2}k_2)$ $k_4\,=\,f(x+h,y+hk_3)$

More sophisticated methods are used by most software programs that solve ODEs, but the concept is the same.