General Solution of the for a Differential Equation in Matrix Form

Homogeneous Solution

Compact vector form

$$\boxed{q_h(t)=VD(t)c}$$

where,

• $V$ is the Matrix with eigen vectors of $A$ $[{\text{q}}^{\prime }(t)=A\text{q}]$
• $D(t)$ is a diagonal matrix with entries $e^{{\lambda }_{1}t},e^{{\lambda }_{2}t},\dots ,e^{{\lambda }_{n}t}$
• $c$ is a vector of the constants $c_1,c_2,\dots ,c_n$.

Long form

$$\boxed{{\mathbf{q}}_{\mathrm{h}}(t)={\mathrm{e}}^{{\lambda }_{1}t}{\mathbf{v}}_1{c}_1+\dots {\mathrm{e}}^{{\lambda }_{n}t}{\mathbf{v}}_n{c}_n}$$