ODE

Example 1

Let us consider the linear differential equation with polynomial coefficients $$ \begin{cases} \displaystyle x\frac{d^2}{dx^2}y+2\frac{d}{dx}y-(6x+4x^3)y=0,\quad x\in[0,\ 1]\\ y(0)=1,\quad y'(0)=0 \end{cases}. $$ The approximate solution on Tau Toolbox:

The approximate tau solution achieves the machine precision: Problems similar with this one can be performed with more specyfic details (intermediate user):

Example 2

Let us consider the linear differential problem $$ \begin{cases} \displaystyle \frac{d^2}{dx^2}y+y=0,\quad x\in[0,\ \pi]\\ y(0)=1,\quad y'(10\pi)=0 \end{cases}. $$ The approximate solution on Tau Toolbox:

The approximate tau solution achieves the machine precision:
Example 3

Let us consider the linear differential problem with non-polynomial coefficients $$ \begin{cases} \displaystyle 10^{-7}\frac{d^2}{dx^2}y-2x(\cos(x)-0.8)\frac{d}{dx}y+(\cos(x)-0.8)y=0,\quad x\in[-1,\ 1]\\ y(-1)=1,\quad y(1)=1 \end{cases}. $$ The approximate solution on Tau Toolbox:

The approximate tau solution is found in 50 pieces: