Mathematics for Electrical Engineering and Computing

14.6 Solving systems of differential equations

To solve a system of differential equations we can combine the equations into a single differential equation using substitution. In this case we can use the method as outlined in Section 14.5. Alternatively, we can solve the system directly using matrices.

We follow the same pattern as in Section 14.4 for first-order systems. We solve the homogeneous equation, to find the complementary function and then find a particular solution. The sum of these two terms gives a general solution to the system.

Example 14.10

Find the displacement of the spring after time t described by the system of differential equations

$\frac{\text{d}}{\text{d}t}\left(\begin{array}{l}{x}_1\hfill \\ x_2\hfill \end{array}\right)=\left(\begin{array}{ll}0\hfill & 1\hfill \\ -k/m\hfill & -r/m\hfill \end{array}\right)?\left(\begin{array}{l}{x}_1\hfill \\ x_2\hfill \end{array}\right)?\left(\begin{array}{l}0\hfill \\ 1/m\hfill \end{array}\right)?f(t)$

in the case where the mass on the spring is 1, the damping ...