Variation of Constants for System of Linear Differential Equation

Homogeneous Solution

q_h(t)=V D(t) c

Particular Solution

q_p(t)=V D(t) c(t) \begin{aligned} q'_p(t)&= V D'(t) c(t)+V D(t) c'(t)\\ &=V \Lambda D'(t) c(t)+V D(t) c'(t)\end{aligned}

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(t)-V \Lambda D'(t) c(t)=V D(t) c'(t) $$\Rightarrow VD(t)c^{\prime }(t)=P(t)$$

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

Solution

$$q=q_h+q_p$$