Understanding Separable Differential Equations

In calculus, a differential equation is an equation that relates an unknown function and its derivatives. A first-order differential equation involves the first derivative of the unknown function. One useful technique for solving these equations is separation of variables.

Definition: A separable differential equation is a first-order differential equation that can be written in the form: [ \frac{dy}{dx} = f(x)g(y) ] where $\frac{dy}{dx}$ represents the derivative of the unknown function $y$ with respect to $x$, and $f(x)$ and $g(y)$ are functions of $x$ and $y$ respectively.

Importance of Separable Differential Equations: Separable differential equations are significant in calculus because they can be solved by a straightforward technique, making them one of the most accessible types of differential equations to solve. Additionally, many real-world problems can be modeled using separable differential equations, which allows us to make predictions and analyze various scenarios.

Separating Variables: The first step in solving a separable differential equation is to separate the variables by multiplying both sides of the equation by $dx$ and dividing by $g(y)$: [ \frac{1}{g(y)} , dy = f(x) , dx ]

Example: Let's solve the separable differential equation: [ \frac{dy}{dx} = x^2e^y ]

Step 1: Separate the variables [ \frac{1}{e^y} , dy = x^2 , dx ]

Step 2: Integrate both sides of the equation [ \int \frac{1}{e^y} , dy = \int x^2 , dx ]

Step 3: Evaluate the integrals Using the fact that $\int e^u \, du = e^u + C$, we have: [ -e^{-y} = \frac{1}{3}x^3 + C ]

where $C$ is the constant of integration.

Step 4: Solve for $y$ [ e^{-y} = -\frac{1}{3}x^3 - C ]

Taking the natural logarithm of both sides gives: [ -y = \ln\left(-\frac{1}{3}x^3 - C\right) ]

Finally, solving for $y$ gives: [ y = -\ln\left(-\frac{1}{3}x^3 - C\right) ]

This is the solution to the given separable differential equation.

By using the technique of separating variables, we were able to solve the separable differential equation and find the general solution. In the next post, we will further explore strategies for solving such equations and work through more examples.