The ordinary differential equation \(\frac{{{\rm{dy}}}}{{{\rm{dt}}}} = - 3{\rm{x}} + {\rm{}}2,{\rm{with\ x}}\left( 0 \right){\rm{\;}} = {\rm{\;}}1\)
Concept:
Euler’s method equation is given by:
yj+1 = yj + h f (xj, yj)
Taking test equation y’ = λy
Yj+1 = yj + h λ yj
= (1 + hλ) yj
Condition for stability is:
|1 + hλ| < 1, i.e.
- 1 < 1 + h λ < 1
Application of Concept:
Given:
\(\frac{{dy}}{{dt}} = - 3x + 2\) with x(0) = 1
f(t, x) = -3x +2
yn+1 = yn + hf (t, yn)
= yn + h(-3yn + 2)
= yn – 3yn h + 2h
Yn+1 = yn (1 – 3h) + 2h
|1 – 3h| < 1
-1 < 1 – 3h < 1
-2 < -3h < 0
0 < 3h < 2
\(0 < h < \frac{2}{3}\)
\({h_{max}} = \frac{2}{3} = 0.66\)
∴ The largest time step that can be used to solve the equation without instability is
hmax = 0.66
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