91Ó°ÊÓ

The concept of separable differential equations is fundamental in solving many ordinary differential equations. A separable differential equation is one where the variables can be separated on either side of the equation. This means you can rearrange the equation so that all terms involving one variable are on one side, and all terms involving the other variable are on the other side. For example, in the differential equation \( (x^2 + 1) y' = xy \), by dividing both sides appropriately, we can express it as \( \frac{1}{y} \, dy = \frac{x}{x^2 + 1} \, dx \). This separation allows us to integrate each side with respect to its respective variable.

Separable differential equations are often the first type of differential equations students learn to solve because they require simple algebraic manipulations and straightforward integration. The key step is identifying if the equation can be "separated" by multiplying or dividing by appropriate expressions. Once separated, solving the equation becomes a matter of integrating each side separately.

Recognizing separability is crucial. If an equation is not separable, more advanced techniques must be used. But when it is, the path to a solution can be quite fast.

After separating the variables in a separable differential equation, the next logical step is integrating both sides. This involves finding the antiderivative, or integral, of each side with respect to its variable. Integrating both sides transforms the differential equation into a form that is easier to solve.

To illustrate, consider the equation \( \frac{1}{y} \, dy = \frac{x}{x^2 + 1} \, dx \). The left side, \( \int \frac{1}{y} \, dy \), integrates to \( \ln|y| + C_1 \). The right side requires substitution. By letting \( u = x^2 + 1 \), we simplify the integration into \( \int \frac{1}{u} \, du = \ln|u| + C_2 \). Substituting back gives \( \frac{1}{2} \ln|x^2 + 1| + C_2 \).

Meshing these integrals together provides us with an equation like \( \ln|y| = \frac{1}{2} \ln|x^2 + 1| + C \), where \( C \) combines the constants from both integrals. This step is about eliminating derivatives, allowing us to focus solely on algebraic equations.

Once integration is complete, the final objective is to find the general solution to the differential equation. A general solution includes an arbitrary constant and represents a family of functions that satisfy the equation. This reflects the nature of differential equations, which often describe continuous processes and phenomena allowing multiple solutions.

In our exercise, after integrating and simplifying, we reach an equation of the form \( \ln|y| = \frac{1}{2} \ln|x^2 + 1| + C \). Exponentiating both sides removes the logarithm, resulting in \( |y| = e^C \sqrt{x^2 + 1} \). To express \( y \) without its absolute value, we introduce a constant \( A = e^C \), giving us the solutions \( y = A \sqrt{x^2 + 1} \) and \( y = -A \sqrt{x^2 + 1} \).

Since \( A \) can be any real number, our general solution \( y = C \sqrt{x^2 + 1} \) (where \( C \) encapsulates the +/- and any numerical constant) covers all possible scenarios. It's a comprehensive solution that accounts for all boundary conditions or initial values that might be applied later.