91Ó°ÊÓ

Separable differential equations are a special type of differential equation where you can isolate each variable on different sides of the equation. To identify if an equation is separable, check if you can write it in the form \( g(y) \, dy = h(x) \, dx \). This means separating the variables into a function of \( y \) times \( dy \) on one side and a function of \( x \) times \( dx \) on the other side.
For example, consider the differential equation \( y' = x - 2xy \). You can factor the right-hand side as \( x(1 - 2y) \), which suggests a clear separation: \( \frac{1}{1 - 2y} \, dy = x \, dx \).
This separation allows you to deal with each variable independently. By recognizing an equation as separable, you can transform the problem into finding antiderivatives (integrating) to solve for the general solution.

Integration is the process of finding the antiderivative or integral of a function. When dealing with separable differential equations, you'll use integration to find the solution.
In our example, once the variables are separated as \( \frac{1}{1 - 2y} \, dy = x \, dx \), integrating both sides with respect to their variables is the next step. The integral of \( \frac{1}{1-2y} \) with respect to \( y \) results in \( -\frac{1}{2} \ln |1-2y| \), and the integral of \( x \) with respect to \( x \) is \( \frac{x^2}{2} \).
Integration adds an arbitrary constant, typically written as \( C \), which is combined from both sides of the equation to simplify our expressions.

• This constant represents the family of functions that satisfy the differential equation.
• The integration process is crucial as it broadens the understanding that multiple solutions exist.

The general solution of a differential equation encompasses all possible solutions derived from the integration process. In our context, once you've integrated both sides and combined results, you'll solve for \( y \).
Using our example, after integration and simplification, you get to an expression like \( \ln |1-2y| = -x^2 - 2C \). Solving further, you exponentiate to isolate \( y \): \( |1-2y| = e^{-x^2 - 2C} \).
This leads to expressing the constant \( e^{-2C} \) as \( C_1 \), yielding the general solution \( y = \frac{1}{2}(1 \mp C_1 e^{-x^2}) \), where \( C_1 \) is a new, positive constant summarizing arbitrary constants.

• The general solution represents the set of all functions that satisfy the original differential equation.
• This form encapsulates infinite possibilities for \( y \) based on different constants \( C_1 \).