Practice Problems: Separable DEs

Separable differential equations are a fundamental concept in calculus and are widely used in various fields of science and engineering, including physics, biology, and economics. These types of equations involve only first-order derivatives and can be solved by separating the variables and integrating each side with respect to its respective variable. Solving separable differential equations is a powerful tool in understanding dynamic systems and their behavior over time. In this tutorial, we will explore the steps involved in solving separable differential equations using several examples and techniques, including separation of variables and partial fractions.

Separable differential equations are arguably the easiest types of differential equations to solve. These types of DEs can be expressed as the product of two functions by separating the variables (hence the name separable differential equations). A first-order differential equation is said to be a separable differential equation if it can be expressed as follows.

\(\displaystyle\frac{dy}{dx}=f(x)g(y)\)

Notice that the function \(f(x)\) contains only the \(x\) terms, whereas \(g(y)\) is comprised of only the \(y\) terms.