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Finding General Solutions Using Separation of Variables

Benjamin Wright

Benjamin Wright

6 min read

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Study Guide Overview

This study guide covers solving separable differential equations. It explains how to identify these equations (in the form dy/dx = g(x)h(y)), and provides a step-by-step process for solving them using separation of variables and integration. The guide includes practice problems demonstrating the technique and discusses finding general solutions. It also briefly mentions using initial conditions to find particular solutions (covered in a separate guide).

7.6 Finding General Solutions Using Separation of Variables

Welcome back to AP Calculus with Fiveable! This topic focuses on separation of variables in differential equations. We’ve worked through modeling differential equations with slope fields and verifying solutions, so lets take the final step and learn how to find the solution to a differential equation. 🙌


🎯 Solving Differential Equations

The solution to a differential equation is any continuous function that satisfies the differential equation. This solution set can include many equations! We learned how to go from solutions to equations in a previous lesson. To brush up on verifying solutions, check out this Fiveable guide: Verifying Solutions to Differential Equations.

But how can we go from an equation to a solution? Many differential equations, such as 2xy' + y = 3x^2 are in fact difficult to solve complex, and are out of the scope of this course. You’ll likely encounter them again in college! But we can solve other types of differential equations: separable differential equations.


👐 Separable Differential Equations

Separable differential equations are first-order differential equations characterized by having two variables, the independent and dependent variables, that can be integrated separately to give a solution to the differential equation.

They are usually written in the form dydx=g(x)h(y)\frac{dy}{dx}​=g(x)h(y), where g(x)g(x) is a function of xx and h(y)h(y)...

Question 1 of 12

Which of the following differential equations is in a separable form?

dydx=x+y\frac{dy}{dx} = x + y

dydx=xy2\frac{dy}{dx} = xy^2

dydx=x2+y2\frac{dy}{dx} = x^2 + y^2

dydx=xy+1\frac{dy}{dx} = \frac{x}{y+1}