The World of Separable Differential Equations
Separable differential equations are a type of first-order ODE where the variables can be separated on either side of the equation. By rearranging, all terms with one variable go on one side, and all terms with the other variable on the opposite side, allowing for straightforward integration and solution.

Solving separable differential equations:
To solve a separable differential equation, the variables \( y \) and \( x \) are first separated. This is followed by integrating both sides of the equation independently. After integration, an arbitrary constant is introduced, and the equation is rearranged to derive the general solution, expressing the relationship between \( y \) , \( x \), and the constant \( C \).
Here is an example:
\( \text{Problem: Solve the separable differential equation }\)
\(\frac{dy}{dx} = \frac{y^2 \sin(x)}{e^y}, \text{ where } y \neq 0.\)
\(\text{First, separate the variables } y \text{ and } x:\)
\(e^y dy = y^2 \sin(x) dx.\)
\(\text{Next, integrate both sides of the equation:}\)
\(\int e^y dy = \int y^2 \sin(x) dx.\)
\(\text{Perform the integrations:}\)
\(e^y = -y^2 \cos(x) + g(y),\)
\(\text{where } g(y) \text{ is an arbitrary function of } y.\)
\(\text{Since the left side is a function of } y \text{ only, } g(y) \text{ must be a constant:}\)
\(e^y = -y^2 \cos(x) + C.\)
\(\text{The general solution to the differential equation is:}\)
\(e^y + y^2 \cos(x) = C,\)
\(\text{where } C \text{ is an arbitrary constant.} \)
Here is another problem:
\( \text{Problem: Solve the differential equation }\)
\(\frac{dy}{dx} = \frac{\cos(x) + \sin(y)}{y(1 + x^2)}, \text{ with } y \neq 0 \text{ and } x \neq 0.\)
\(\text{Integrate both sides:}\)
\(\int y(1 + x^2) dy = \int (\cos(x) + \sin(y)) dx.\)
\(\text{Performing the integrations:}\)
\(\frac{y^2}{2} + \frac{y^2 x^2}{2} = \sin(x) – \cos(y) + C,\)
\(\text{where } C \text{ is an integration constant.}\)
\(\text{The general solution to the differential equation is:}\)
\(\frac{y^2}{2} + \frac{y^2 x^2}{2} = \sin(x) – \cos(y) + C,\)
\(\text{expressing a relationship between } y, x, \text{ and } C. \)
Related to This Article
More math articles
- 8th Grade SOL Math Worksheets: FREE & Printable
- Absolute vs Conditional Convergence: Key Differences Explained
- Full-Length 7th Grade SBAC Math Practice Test-Answers and Explanations
- The Ultimate 6th Grade STAAR Math Course (+FREE Worksheets)
- Full-Length 7th Grade FSA Math Practice Test
- ALEKS Math FREE Sample Practice Questions
- How to Use Area Models to Find Equivalent Fractions
- The Ultimate SSAT Middle Level Math Formula Cheat Sheet
- How to Find the Domain and Range of Quadratic Functions
- How to Find Volume by Spinning: Washer Method
What people say about "The World of Separable Differential Equations - Effortless Math: We Help Students Learn to LOVE Mathematics"?
No one replied yet.