Basics

Homogeneous LTI DE #

homogeneous linear time-invariant differential equations

$$ \begin{align*} \frac{dx}{dt}&= F x\\ \text{By separation of variables}\\ x(t)&= e^{Ft}x(0)\\ \\ x(t)&= x(0)+\int^{t} F x(\tau)d \tau\\ x(t)&= x(0)+\int^{t} F\left[x(0)+\int^{t} F x(\tau)d \tau\right]d \tau\\ &= x(0)+Fx(0)t+\iint F^{2}x(\tau)d \tau^{2}\\ &= x(0)(1+Ft+ \frac{F^{2}t^{2}}{2!}+ \frac{F^{3}t^{3}}{3!}+..)\\ &= x(0)e^{Ft} \end{align*} $$

This would work even if F is a matrix!

Inhomogeneous LTI DE #

$$ \begin{align*} \frac{dx}{dt}=Fx(t)+L w(t)\\ \text{Integrating factor: }e^{-Ft}\\ e^{-Ft} \frac{dx}{dt}- e^{-Ft}Fx=e^{-Ft}Lw(t)\\ \frac{d}{dx}(e^{-Ft}x)=e^{-Ft}Lw(t)\\ e^{-Ft}x-e^{-Ft_{0}}x(t_{0})=\int_{t_{0}}^{t}e^{-F \tau}Lw(\tau)d \tau\\ x(t)=e^{Ft-Ft_{0}}x(t_{0})+\int_{t_{0}}^{t}e^{Ft-F \tau}Lw(\tau)d \tau \end{align*} $$

Time variant General DE #

homogeneous #

$$ \begin{align*} \frac{dx}{dt}=F(t)x(t)\\ \\ \text{Assume solution }x(t)=\psi(t,t_{0})x(t_{0}) \end{align*} $$

inhomogeneous #

$\psi(t_{0},t)$ is the integrating factor in this case

$$ \begin{align*} \frac{dx}{dt}=F(t)x(t)+L(t)w(t)\\ x(t)=\psi(t,t_{0})x(t_{0})+\int_{t_{0}}^{t}\psi(t,\tau)Lw(\tau)d \tau \end{align*} $$

Picard Lindelof theorem #

Generic Equation #

$$ \frac{dx}{dt}=f(x(t),t);x(t_{0})=x_{0} $$

Picard Iteration #

Similar to ODE Methods

$$ \begin{align*} \phi_{n+1}(x)=x_{0}+\int_{t_{0}}^{t}f(\phi_{n}(\tau),\tau)d \tau\\ \\ \lim_{n\rightarrow \infty}(\phi_{n}(t))=x(t) \end{align*} $$

provided that it’s continuous in both arguments and lipschitz continuous in first argument.

The theorem #

Under the above continuity conditions the differential equation has a solution and it is unique at a certain interval around $t_{0}$