Solving the ODE exactly
Equation is linear
- Homogeneous, constant coefficients
-
- \(u'(t)=\mathrm{A} u(t)\), \(u(0)=u_0\)
- \(\Rightarrow u(t) = u_0\,\exp(\mathrm{A}\,t)\)
- Non-Homogeneous, constant coefficients
-
- \(u'(t)=\mathrm{A} u(t)+b(t)\), \(u(0)=u_0\)
- \(\Rightarrow \displaystyle u(t) = u_0\,\exp(\mathrm{A}\,t) + \exp\left(\mathrm{A}\,t\right)\int_{t'=0} ^ {t}
b(t')\exp\left(-\mathrm{A}\,t'\right)\,\mathrm{d}t' \)
- Homogeneous, variable coefficients
-
- \(u'(t)=\mathrm{A}(t)\,u(t)\), \(u(0)=u_0\)
- \(\Rightarrow\) No closed-form solution in the general case
-
- - Rem. Unfortunately, in general, \(u(t)\) is not \(\displaystyle u_0\,\exp\left(\int_{0}^{t}
\mathrm{A}(t')\,\mathrm{d}t' \right)\)
Equation is non linear
- \(\Rightarrow\) Numerical approaches are required most of the time