Exercise 1496Calculus › Differential equations › Linear systemsL3Added 2 Nov 2026

A linear system with a second member

#second member#variation of constants#matrix exponential

Statement

Solve the system $X' = AX + B(t)$ where $A$ is diagonalizable and $B(t)$ is a given vector function.

Hint

Use variation of constants: $X(t) = e^{tA}C(t)$ , so $C'(t) = e^{-tA}B(t)$ .

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Solution

Variation of constants. Set $X(t) = e^{tA}C(t)$ . Then
$X' = A e^{tA}C + e^{tA}C' = AX + e^{tA}C'.$
We need $e^{tA}C' = B(t)$ , so
$C'(t) = e^{-tA}B(t).$

Integration.
$C(t) = C_0 + \int_{t_0}^{t} e^{-sA}B(s)\,ds.$

General solution.
$X(t) = e^{tA}C_0 + \int_{t_0}^{t} e^{(t-s)A}B(s)\,ds.$

A linear system with a second member

Statement

Solution

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