# Preface

Stiff

Introduction to Linear Algebra with Mathematica

# Stiff equations

Stiff

Example: Consider the initial value problem

$\begin{cases} \dot{x} &= 520\,x -2605\,y \\ \dot{y} &= 304\,x -1521\, y ; \end{cases}\qquad \begin{bmatrix} x(0) \\ y(0) \end{bmatrix} = \begin{bmatrix} 1 \\ -1 \end{bmatrix}$
The true solution is
\begin{align*} x(t) &= \frac{1}{333} \left[ 1375\, e^{-t} - 1042\, e^{-1000\,t} \right] , \\ y(t) &= \frac{1}{333} \left[ 275\,e^{-t} -608\, e^{-1000\,t} \right] . \end{align*}
DSolve[{x'[t] == 520*x[t] - 2605*y[t], y'[t] == 304*x[t] - 1521*y[t], x[0] == 1, y[0] == -1}, {x, y}, t]
{{x -> Function[{t}, 1/333 E^(-1000 t) (-1042 + 1375 E^(999 t))], y -> Function[{t}, 1/333 E^(-1000 t) (-608 + 275 E^(999 t))]}}
The above system of differential equations can be reduced to one single equation:
$x'' (t) + 1001\, x' (t) + 1000\,x(t) = 0, \qquad x(0) =1, \quad x' (0) = 3125.$
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1. E. Hairer and G. Wanner. Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems. Springer-Verlag, 1991.
2. Abdul-Majid Wazwaz, Partial Differential Equations and Solitary Waves Theory, Nonlinear Physical Science. Springer, Berlin, Heidelberg, 2009. https://doi.org/10.1007/978-3-642-00251-9_5