Systems of ODEs with constant coefficients

We begin with a system of linear differential equations. (Higher order differential equations can be transformed to this form.) A general first-order linear system has the form  

$${d\over dt}X(t)= A X(t)+G(t)$$ (1)

where X(t) and G(t) are $n\times 1$ vectors, A is an $n\times n$ matrix of constant coefficients.

For a homogeneous system,  

$${d\over dt}X(t)= A X(t),$$ (2)

we need n linearly independent solutions to construct the general solution  

$$X(t)=c_1\phi_1(t)+c_2\phi_2(t) +...+c_n\phi_n(t)$$ (3)

where

$$\phi_i(t)=\pmatrix{\phi_{1i}\cr \phi_{2i}\cr ...\cr \phi_{ni} }.$$

We can write the solution as

$$X(t)=\Omega C,$$ (4)

where

$$\Omega=\pmatrix{\phi_1 & \phi_2 & ... & \phi_n}$$

is a matrix with the solutions as columns and

$$C=\pmatrix{c_1 & c_2 & ... & c_n}^T.$$

The matrix $\Omega$ is a fundamental matrix of the system $\dot X=AX$. Note

$$\Omega C=\pmatrix{\phi_1 & \phi_2 & ... & \phi_n} \pmatrix{c_1 \cr c_2 \cr ... \cr c_n}$$

$=c_1\phi_1+c_2\phi_2 + ... + c_n\phi_n$, which is the general solution.

Example 1

$$\dot x_1=x_1-4x_2,\quad \dot x_2=x_1+5x_2$$

$$\to \pmatrix{\dot x_1\cr \dot x_2}=\pmatrix{1 & -4\cr 1 & 5\cr} \pmatrix{x_1\cr x_2}.$$

We can check by substitution that two linearly independent solutions are

$$\phi_1=\pmatrix{-2e^{3t} \cr e^{3t} },\quad \phi_2=\pmatrix{(1-2t)e^{3t}\cr te^{3t}\cr}.$$

Thus, the general solution is

$$\pmatrix{x_1\cr x_2}=\pmatrix{ -2e^{3t} & (1-2t)e^{3t}\cr e^{3t} & te^{3t}\cr}\pmatrix{c_1\cr c_2}$$

$$=\pmatrix{ -2c_1e^{3t}+c_2 (1-2t)e^{3t}\cr c_1e^{3t} +c_2te^{3t}} .$$

Recall that: If C is any nonsingular matrix ( $\det C \ne 0$) with constant entries, then $\Omega C=\Psi$ is another fundamental matrix. Check this: $d\Psi/dt=d(\Omega C)/dt=A\Omega C= A\Psi$ and $\det\Psi= \det(\Omega C) = \det\Omega \det C \ne 0$.