Practice Problems: Integrating Factor

Differential equations arise in many areas of science and engineering, including physics, chemistry, and biology. Solving differential equations using the integral factor is a powerful technique that can be used to solve first-order linear ordinary differential equations with constant coefficients. This technique involves multiplying both sides of the differential equation by an integrating factor that is derived from the coefficient of the dependent variable. In this tutorial, we will explore the steps involved in solving differential equations using the integral factor, including finding the integrating factor and solving for the general solution.

The general formula for a first-order linear ordinary differential equation with constant coefficients is: \(\frac{dy}{dx} + p(x)y = q(x)\) where \(p(x)\) and \(q(x)\) are continuous functions of \(x\). To solve this differential equation using the integral factor, we need to find an integrating factor \(u(x)=e^{\int p(x)dx}\) such that: \(u(x) \frac{dy}{dx} + p(x)u(x) y = u(x)q(x)\). This equation can be written in the product rule form \(\frac{d}{dx} (u(x)y) = u(x)q(x)\)

By integrating both sides with respect to \(x\), we can solve for \(u(x)\) and find the general solution to the differential equation. This technique is particularly useful for solving differential equations with non-constant coefficients.