We know that complex numbers can be written as: $$a+ib\equiv\begin{pmatrix}a&-b\\ b& a \end{pmatrix}$$
in matrix form. Suppose we want to solve the differential equation: $$f'(x)+if(x)=0$$ The solution is $f(x)=c e^{-ix}$. Suppose that instead, we want to solve this equation by writing the above equation in a matrix form, i.e. $$f'(x)+if(x)\equiv \begin{pmatrix}f'(x)&-f(x)\\ f(x)& f'(x) \end{pmatrix}$$ My question is: Is it possible to solve this equation and, in general, other possibly non-linear ODE's using the matrix representation?
Thanks, Michał
Both $f(x)$ and $f'(x)$ are also complex numbers. So if $f(x)=g(x)+ih(x)$, then you can write $$f(x)=\begin{bmatrix}g(x)&-h(x)\\h(x)&g(x)\end{bmatrix},$$ so your differential equation becomes $$\begin{bmatrix}g'(x)&-h'(x)\\h'(x)&g'(x)\end{bmatrix}+\begin{bmatrix}0&-1\\1&0\end{bmatrix}\begin{bmatrix}g(x)&-h(x)\\h(x)&g(x)\end{bmatrix}=\begin{bmatrix}0&0\\0&0\end{bmatrix}.$$ You can solve this; it is a coupled set of differential equations $$g'(x)-h(x)=0; h'(x)+g(x)=0,$$ which can become a second order differential equation: $$g''(x)+g(x)=0$$