Variation of Constants for System of Linear Differential Equation

Homogeneous Solution

$$q_h(t)=VD(t)c$$

Particular Solution

$$q_p(t)=VD(t)c(t)$$ $$\begin{array}{rl}{q}_p^{\prime }(t)& =V{D}^{\prime }(t)c(t)+VD(t)c^{\prime }(t)\\ & =V\Lambda D^{\prime }(t)c(t)+VD(t)c^{\prime }(t)\end{array}$$

Note: $D(t)$ is a diagonal matrix with entries $e^{{\lambda }_{1}t},e^{{\lambda }_{2}t},\dots ,e^{{\lambda }_{n}t}$ . Through differentiation $\lambda$ gets multiplied for each element. It can be conveniently represented by $\Lambda$ where each diagonal element is an eigen value.

Similar to the 1-D case,

$$q_p^{\prime }(t)-V\Lambda D^{\prime }(t)c(t)=VD(t)c^{\prime }(t)$$ $$\Rightarrow VD(t)c^{\prime }(t)=P(t)$$

Integrate each element of to get $c(t)$

Solution

$$q=q_h+q_p$$