First Order Linear Differential Equations

The first order linear ODE is a powerful class of solvable ODEs (DEs). They are defined by the following two equivalent terms.

\[ A(t) \frac{dx}{dt} + B(t)x(t) = C(t) \]

Or if we factor out the \( A(t) \):

\[ \frac{dx}{dt} + p(t)x(t) = q(t) \]

Where \(p(t) = \frac{B(t)}{A(t)} \).

Of note, we consider the equation 'homogenous' if \( C(t) = 0 \).