91Ó°ÊÓ

To tackle a differential equation like the one given, we often use the method of separation of variables. This technique simplifies solving differential equations when variables can be isolated onto different sides of the equation.
The primary goal of this method is to rewrite the differential equation such that all terms involving the dependent variable (in our case, \(y\)) are on one side and all terms involving the independent variable (\(x\)) are on the other side.

• First, identify the parts of the equation that relate to \(y\) and \(x\).
• Rewrite the equation so that the \(dy\) terms are on one side and the \(dx\) terms are on the other.

In the given problem, we started with the differential equation \( \frac{dy}{dx} = (1+y^2)x^2 \). By rewriting it as \( dy = (1+y^2)x^2 dx \) and then dividing by \(1+y^2\), the equation becomes \( \frac{dy}{1+y^2} = x^2 dx \). This separation of variables allows each side of the equation to be integrated individually.

Once the variables are separated, we proceed to integrate both sides. Integration is a mathematical process that helps us find the function from its derivative.
The idea is to calculate an antiderivative or the original function from the rate of change, which we have expressed with differentials.

• The left side: \( \int \frac{dy}{1+y^2} \), involves a known integral resulting in \( \arctan(y) \).
• The right side: \( \int x^2 dx \), results after calculation in \( \frac{x^3}{3} + C \), where \(C\) is the constant of integration.

After integrating both sides, they equal to \( \arctan(y) = \frac{x^3}{3} + C \). The integration helps us transition from a relationship in terms of derivatives to a function that describes \(y\) in terms of \(x\). Remember, it is important always to include the constant \(C\) as it represents an infinite number of possible solutions.

The family of solutions in a differential equation context refers to the infinite set of solutions that satisfy the differential equation. This family arises due to the constant of integration, \(C\), introduced during the integration process.
Each unique value of \(C\) corresponds to a different solution, representing a particular function that satisfies the original equation.

• From our integrated equation: \( \arctan(y) = \frac{x^3}{3} + C \), solving for \(y\) reveals: \( y = \tan\left( \frac{x^3}{3} + C \right) \).
• This solution is explicit because it directly expresses \(y\) in terms of \(x\), with \(C\) being any real number, showcasing an entire set of potential solutions.

Different values of \(C\) will adjust the angle inside the tangent function, creating a progression of curves that are all valid solutions to the equation. Thus, the concept of a 'family of solutions' is crucial as it underlines that the solution to differential equations is not just a single function, but a collection of them, influenced by initial conditions or constraints.