IDE

Example 1

Let us consider the linear integro-differential equation $$ \begin{cases} \displaystyle \displaystyle\cos(x)\frac{d^2}{dx^2}y - \sin(x^3-x)\frac{d}{d}y+\exp({\frac{x^2}{2}})y - \cosh(x)\int_{0}^{x}\sin(x-4t)y(t)dt \\\hfill \displaystyle+\sinh(x)\int_{0}^{1}\cos(t^2-x)y(t)dt = \exp({\sin(x^3-x)+x^2})\\ y(0)=1,\ y(1)=0 \end{cases}. $$ The approximate solution on Tau Toolbox:

The approximate tau solution achieves the machine precision:
Example 2

Let us consider the linear integro-differential equation $$ \begin{cases} \displaystyle \frac{d}{dx}y+2y+5\int_{0}^{x}y(t)dt=1\\ y(0)=0 \end{cases}. $$ The approximate solution on Tau Toolbox:

The approximate tau solution achieves the machine precision: