Linear Differential Equations

General Form of a Linear Differential Equation

The general form of a linear, first-order differential equation is that we have a differential term, a function multiplied by y, and then some forcing function g(t).

The Integrating Factor

Normally, linear differential equations cannot be easily solved in their current form; however, if we introduce an integrating factor, then we can easily solve for its solution.

The integrating factor is always the exponent of the integral of the middle term.

Process of Solving

Once we have identified the integrating factor, we then multiply the entire equation by it; from there, we then collapse the left-hand side into a single differential via the product rule as necessary.

From there, we then integrate both sides and obtain our y(t).

Example of Solving a Linear-Differential Equation

Notice how the expression is reduced to a single differential inside the parantheses; from there, we integrate as necessary to obtain y(t).