91Ó°ÊÓ

In differential equations, a separable equation is one that can be rewritten to allow the separation of variables. This means you can express the equation such that each variable appears on its own side of the equation with its respective differential. For example, the equation given here is \( y' = x - 2xy \). Our goal is to rewrite it and check if it can be separated.Notice how the equation can be reorganized into \( y' = x(1 - 2y) \). By doing this, we reveal a form that allows separation: \( \frac{dy}{1 - 2y} = x \, dx \). This split means that each variable (\(y\) and \(x\)) appears only with its differential (\(dy\) or \(dx\), respectively).Key steps to identify a separable equation:

• Rearrange terms to isolate multiplication or division forms.
• Observe if variables can be separated on opposite sides of the equation.
• Check if both sides have a consistent and meaningful differential (\(dy\) and \(dx\)).

Once you have successfully separated a differential equation, it’s time to integrate both sides. Integration allows you to solve for the function that the differential equation describes.In this scenario, the separated equation \( \frac{dy}{1 - 2y} = x \, dx \) requires us to integrate each side:- The left side becomes \( \int \frac{dy}{1 - 2y} \).- The right side becomes \( \int x \, dx \).The integration process:

• For the left side, you recognize the form of the integral as \(-\frac{1}{2} \ln |1 - 2y| \).
• For the right side, the integral is straightforward, \( \frac{x^2}{2} + C \), with \( C \) as the integration constant.

These results offer expressions for each integral that come together to give a more complete picture of the solution.

The general solution is a key outcome when solving differential equations, representing a family of solutions rather than a single one. After integrating the separated equation, you reach an expression like \(-\frac{1}{2} \ln |1 - 2y| = \frac{x^2}{2} + C \).The steps to find the general solution are:

• Eliminate the logarithm by exponents to solve for the function \(y\).
• Result in the simplified form \(|1 - 2y| = e^{-x^2 - 2C}\). Substitute \(e^{-2C}\) by a constant \(K\), yielding \(1 - 2y = \pm Ke^{-x^2}\).
• Finally, isolate \(y\) so that \(y = \frac{1 \mp Ke^{-x^2}}{2}\).

This general solution contains arbitrary constants indicating an infinite number of possible specific solutions, depending on initial conditions. Solver can check if the obtained expression satisfies the original equation and aligns with any given initial conditions in a more specific problem scenario.