Résoudre x=x_{0}e^rt

$$x = x _ { 0 } e ^ { r t }$$

Calculer r (solution complexe)

$$\left\{\begin{matrix}r=\frac{2\pi n_{1}i}{t}+\frac{\ln(\frac{x}{x_{0}})}{t}\text{, }n_{1}\in \mathrm{Z}\text{, }&t\neq 0\text{ and }x_{0}\neq 0\text{ and }x\neq 0\\r\in \mathrm{C}\text{, }&\left(x=0\text{ and }x_{0}=0\right)\text{ or }\left(x_{0}=x\text{ and }t=0\text{ and }x\neq 0\right)\end{matrix}\right,$$

Calculer t (solution complexe)

$$\left\{\begin{matrix}t=\frac{2\pi n_{1}i}{r}+\frac{\ln(\frac{x}{x_{0}})}{r}\text{, }n_{1}\in \mathrm{Z}\text{, }&r\neq 0\text{ and }x_{0}\neq 0\text{ and }x\neq 0\\t\in \mathrm{C}\text{, }&\left(x=0\text{ and }x_{0}=0\right)\text{ or }\left(x_{0}=x\text{ and }r=0\text{ and }x\neq 0\right)\end{matrix}\right,$$

Calculer r

$$\left\{\begin{matrix}r=\frac{\ln(\frac{x}{x_{0}})}{t}\text{, }&\left(t\neq 0\text{ and }x>0\text{ and }x_{0}>0\right)\text{ or }\left(t\neq 0\text{ and }x<0\text{ and }x_{0}<0\right)\\r\in \mathrm{R}\text{, }&\left(x=0\text{ and }x_{0}=0\right)\text{ or }\left(x_{0}=x\text{ and }t=0\text{ and }x\neq 0\right)\end{matrix}\right,$$

Calculer t

$$\left\{\begin{matrix}t=\frac{\ln(\frac{x}{x_{0}})}{r}\text{, }&\left(r\neq 0\text{ and }x>0\text{ and }x_{0}>0\right)\text{ or }\left(r\neq 0\text{ and }x<0\text{ and }x_{0}<0\right)\\t\in \mathrm{R}\text{, }&\left(x=0\text{ and }x_{0}=0\right)\text{ or }\left(x_{0}=x\text{ and }r=0\text{ and }x\neq 0\right)\end{matrix}\right,$$