next up previous contents
Next: Laplace transform. Up: Ordinary differential equations. Previous: Ordinary differential equations.

   
Linear differential equations (systems) with constant coefficients.

The most efficient tool for these equations is the Laplace transform defined by:

\begin{displaymath}Y(s)={\cal L}(y)(s)=\int _0^{\infty} e^{-st} y(t) \ dt \end{displaymath}

Example: solve $y'+2y=\cos(x)$. Apply ${\cal L}$, since:

\begin{displaymath}{\cal L}(y')(s)=s {\cal L}(y)(s)-y(0) \end{displaymath}

we get:

\begin{displaymath}(s+2){\cal L}(y)(s)={\cal L}(\cos(x))(s)+y(0) \end{displaymath}

hence:

\begin{displaymath}y(x)={\cal L}^{-1}\left(\frac{\frac{s}{s^2+1}+y(0) }{s+2}\rig...
...\frac{s}{s^2+1} }{s+2}\right)+y(0){\cal L}^{-1}(\frac{1}{s+2}) \end{displaymath}

since ${\cal L}(\cos(x))(s)=s/(s^2+1)$. This method is implemented by the LDEC instruction. It takes the second member at level 2 (here $\cos(x)$), and the characteristic equation at level 1 (here x+2) and returns the solution vanishing at the origin at level 1, the characteristic equation at level 2, and 1 at level 3. Be sure that the current variable name contained in VX is the right one!



 

Bernard Parisse
1998-07-31