- Calculus, Volume 1: One Variable Calculus, with an Introduction to Linear Algebra
- Tom M. Apostol
- Second Edition
- 1967
- 978-1-119-49673-1
14.4 Exercises
- A differential equation of the form $Y'(x) + p(x)Y(x) = Q(x),$ where $p$ is a given real-valued function, $Q$ a given vector-valued function, and $Y$ an unknown vector-valued function, is called a first-order linear vector differential equation. Prove that if $p$ and $Q$ are continuous on an interval $I,$ then for each $a$ in $I$ and each vector $B$ there is one and only one solution $Y$ which satisfies the initial condition $Y(a) = B,$ and that this solution is given by the formula: $$ \begin{align*} \\ Y(t) = Be^{-q(t)} + e^{-q(t)}\int_a^t Q(x)e^{q(x)}\,dx \end{align*} $$ where $q(x) = \int_a^x p(t)\,dt.$
- We can apply the result of Theorem 8.3 component-wise to achieve this result for a vector-valued function.