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 \\).