Problem 19 - Calculus Early Transcendentals

Question

Solve the differential equation. If you have a CAS with implicit plotting capability, use the CAS to generate five integral curves for the equation. $$ y^{\prime}=\frac{x^{2}}{1-y^{2}} $$

Step-by-Step Solution

Answer

Separate variables, integrate; implicit curves: \( y - \frac{y^3}{3} = \frac{x^3}{3} + C \).

Step 1: Separate the variables

To solve the differential equation \( y' = \frac{x^2}{1-y^2} \), the first step is to separate the variables. This means rewriting the equation so that all terms containing \( y \) are on one side and all terms containing \( x \) are on the other side. Thus, we rewrite it as \( (1-y^2)dy = x^2 dx \).

Step 2: Integrate both sides

Now, integrate both sides of the separated equation. The left side becomes \( \int (1-y^2) \, dy \) and the right side becomes \( \int x^2 \, dx \). After integrating, we get \( y - \frac{y^3}{3} = \frac{x^3}{3} + C \), where \( C \) is the constant of integration.

Step 3: Rewrite the solution

Rewrite the integrated equation in a more explicit form if needed. In this case, \( y - \frac{y^3}{3} = \frac{x^3}{3} + C \) is already an implicit solution for \( y \).

Step 4: Plot the integral curves using CAS

Use a Computer Algebra System (CAS) to plot the integral curves. Choose different values for the constant \( C \) to generate different curves. For instance, you can plot curves for \( C = -2, -1, 0, 1, 2 \) to see how the integral curves vary.

Key Concepts

Separation of Variables

Separation of variables is a powerful technique used to solve differential equations, specifically those that can be expressed as products of functions of the dependent and independent variables. The core idea is to move all terms involving the dependent variable (in this case, "y") to one side of the equation and all terms involving the independent variable ("x") to the other side. This creates two separate integrals that can be solved independently.

• In the given exercise, the differential equation is initially presented as \( y' = \frac{x^2}{1-y^2} \).
• By multiplying both sides by \( (1-y^2) \) and dividing by \( x^2 \), we achieve the separation: \( (1-y^2)dy = x^2 dx \).

The transformation into separate integrals is crucial, as it allows us to integrate each side with respect to its variable, thus simplifying the process of finding a solution. It is a fundamental step that guarantees the possibility to move forward in solving the differential equation.

Integral Curves

Integral curves represent the solution sets of a differential equation in the plane of the independent and dependent variables. These curves help us visualize how solutions behave and vary with different initial conditions. They give a graphical insight into the nature of the solutions and the influence of different constants of integration.

• In this process, once we have the implicit solution \( y - \frac{y^3}{3} = \frac{x^3}{3} + C \), we can consider different values of \( C \) to plot various integral curves.
• For our example, values such as \( C = -2, -1, 0, 1, 2 \) can be used to appreciate how solutions change with different constants.

These curves are essential for understanding the broader spectrum of the differential equation's solutions. They essentially "paint" the landscape of the differential equation, showing how solutions flow and differ across the plane.

Implicit Solution

An implicit solution to a differential equation is one where the dependent variable is not isolated on one side of the equation. Instead, it remains intermingled with the independent variable and possibly other constants or functions. Implicit solutions can often be more convenient to work with, especially when isolating the dependent variable is complex or impossible.

• In the original exercise, the integration process yields the equation \( y - \frac{y^3}{3} = \frac{x^3}{3} + C \), which stands as an implicit solution.
• Here, "y" is not isolated due to the cubic term, and this form may encapsulate more intricate relationships between "x" and "y" that are otherwise lost in attempting an explicit form.

Implicit solutions play a crucial role in mathematical analysis and modeling, as they oftentimes offer more flexibility and capture the interrelationships within a system of equations. They provide a meaningful way to express solutions when explicit forms are either too cumbersome or unavailable.