First-order differential equations

First-order linear equations

General form

A first-order linear differential equation is written as: y' + a(x) * y = b(x) where a and b are continuous functions on an interval I. The case where b(x) = 0 is called homogeneous (or without a right-hand side).

Solving the homogeneous equation

We first solve y' + a(x) * y = 0, which is separable: y'/y = -a(x) -> ln|y| = -A(x) + C, where A is an antiderivative of a Hence y_h(x) = K * exp(-A(x)), where K is a real number.

Method of Variation of a Constant

To find a particular solution to the complete equation, we set y(x) = K(x) * exp(-A(x)), where K(x) becomes a function. Substituting this into the equation, the term K * y_h cancels out (since y_h is a solution to the homogeneous equation), leaving: K'(x) = b(x) * exp(A(x)) We integrate to obtain K(x), then y(x).

Example

Solve y' + 2y = e^(-x). Homogeneous equation: y_h = K * exp(-2x). Differentiation: $K'(x) = e^{-x} \cdot e^{2x} = e^x$ , so $K(x) = e^x + C$ . General solution: y(x) = (e^x + C) * e^(-2x) = e^(-x) + C * e^(-2x)

Summary table

Step	Action
1	Solve the associated homogeneous equation
2	Apply the variation of the constant
3	Integrate K'(x) to obtain K(x)
4	Write y = K(x) * y_h(x)