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.