Question

I need to solve this equation in Mathematica:

d/dx v(x) = A . v(x)

here v is the column vector {v1(x),v2(x),v3(x),v4(x)} and
A is a 4 x 4 matrix.
I want to solve for the functions v1, v2, v3, v4 with any initial conditions.
The range of x is from 0 to 1000.

How do I write Mathematica code for this type of differential equation using NDSolve?

Answer 1

So, if you have some horrible matrix

A =  RandomReal[0.1, {4, 4}]; (* A horrible matrix *)

which we make anti-symmetric (so the solution is oscillatory)

A = A - Transpose@A;

Define the vector of functions and their initial conditions

v[x_] := {v1[x], v2[x], v3[x], v4[x]};

init = v[0] == RandomReal[1, 4]

Then the NDSolve command looks like

sol = NDSolve[LogicalExpand[v'[x] == A.v[x] && init], 
        {v1, v2, v3, v4}, {x, 0, 1000}]

And the solutions can be plotted with

Plot[Evaluate[v[x] /. sol], {x, 0, 1000}]

---

Note that that the above differential equation is a linear, first order equation with constant coefficients, so is simply solved using a matrix exponential.
However, if the matrix A was a function of x, then analytic solutions become hard, but the numerical code stays the same.

For example, try:

A = RandomReal[1/10, {4, 4}] - Exp[-RandomReal[1/100, {4, 4}] x^2];
A = A - Transpose@A;

Which can produce solutions like