[[Ordinary Differential Equations (ODEs)|ODE]]: $ \dot{x}=ax\Longrightarrow x(t)=e^{at}x(0) $ Use [[Taylor Series|Taylor series]]: $x(t)=c_0+c_1t+c_2t^2+\dots\underbrace{\mathcal{O}(t^3)}_{\substack{\text{higher-order}\\\text{terms}}}$ Take [[Derivative|derivative]]: $ \dot{x}(t)=0+c_1+2c_2t+3c_3t+4c_4t^4+\dots\underbrace{\mathcal{O}(t^4)}_{\substack{\text{higher-order}\\\text{terms}}} $ Going term-by-term by powers of t: $ \begin{align} t^0:&&c_1=ac_0 \\ t^1:&&2c_2=ac_1 \\ t^2:&&3c_3=ac_2\\ t^3:&&4c_4=ac_3\\ t^n:&&nc_n=ac_{n-1} \end{align} $ Use initial condition $x(0)=x_0$: $ \begin{align} t^0:&&c_1&=ac_0=ax_0 \\ t^1:&&2c_2&=ac_1\longrightarrow c_2=\frac{a^2x_0}{2}\\ t^2:&&3c_3&=ac_2\longrightarrow c_3=\frac{a^3x_0}{3!}\\ t^n:&&nc_n&=ac_{n-1}\longrightarrow c_n=\frac{a^nx_0}{n!} \end{align} $ Plug into Taylor series expansion: $ \begin{align} x(t)&=x_0+ax_0\,t+\frac{a^2x_0}{2}t^2+\dots\\ &=x_0\left(1+at+\frac{(at)^2}{2!}+\dots\right)\\ &=\boxed{x_0e^{at}} \end{align} $ Extend to matrices: $ \dot{\mathbf{x}}=\mathbf{Ax}\Longrightarrow\mathbf{x}(t)=e^{\mathbf{A}t}\mathbf{x}(0) $ Is basis of all numerical approximations in this course Discretize: A function $f(x+\Delta x)$ can be Taylor expanded at a point $x$ $ f\left(x+\Delta x\right)=f(x)+f'(x)\Delta x+f''(x)\Delta x^2 + \dots $ Can vary $x$ and $\Delta x$