91Ó°ÊÓ

Separation of variables is a fundamental technique used to solve differential equations. This method involves rearranging the given equation to isolate each variable on a separate side, essentially splitting the equation into two simpler integrals. In our exercise, we began with the differential equation \((1+y^2)y' = e^x y\). By rearranging, we aim to express all terms with \(y\) on one side and terms with \(x\) on the other.
This required dividing both sides of the equation by \(y(1+y^2)\), resulting in \(\frac{y'}{y} = \frac{e^x}{1+y^2}\).
The ultimate goal is to get into a form that allows us to integrate each side independently, leading us towards the solution of the differential equation.

Once the variables have been separated, the next step is to integrate both sides of the resulting equation. We focus on each variable individually. This means integrating the left side with respect to \(y\) and the right side with regard to \(x\).
For the left side of the equation, we have \(\int \frac{dy}{y}\), which integrates to \(\ln |y|\).
The right side involves a more complex integral, \(\int e^x \frac{dx}{1+y^2}\). To tackle this, we actually perform an indirect step here by realizing the simpler integration could refer specifically to \(\int e^x dx\). Each integration step provides a component of the potential solution, coalescing into a complete form of the integral equation.

In solving differential equations, finding a family of solutions is crucial since these equations often do not have a single answer. Instead, they offer a set, or 'family' of solutions encompassing a general form that describes all possible solutions based on an arbitrary constant.
The integration ends with \(\ln |y| = e^x + C_1\). To solve for \(y\), we exponentiate both sides to remove the natural log and achieve \(y = \pm e^{e^x + C_1}\).
By introducing a new constant \(C\) where \(e^{C_1} = C\), we simplify our family to \(y = Ce^{e^x}\). This expression represents all potential solutions, with different \(C\) values defining distinct solution curves.

The substitution method is another helpful technique often used to simplify complex integrals within differential equations. Here, substitution helps to break down parts of the integration process by introducing a new variable to represent an existing part of the expression. Although not directly needed in the main solution flow here, substitution is instrumental in similar contexts.
In embedded problems like \(\int \frac{e^x}{1+y^2} dx\), a substitution might typically involve setting \(u = \) some part of the expression that simplifies the integral. The key advantage of substitution is that it transforms difficult integrals into manageable ones, allowing for easier completion of the solution.
While we didn’t explicitly need a substitution to conclude this particular problem, it’s a valuable tool in a mathematician's problem-solving toolkit for more intricate integrals.