Problem 20 - 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. $$ y^{\prime}=\frac{y}{1+y^{2}} $$

Step-by-Step Solution

Answer

The general solution is \( \ln|y| + \frac{y^2}{2} = x + C \). Use a CAS to plot integral curves for different \( C \).

Step 1: Understand the Differential Equation

The given differential equation is \( y' = \frac{y}{1+y^2} \). We recognize this as a separable differential equation, which means we can express it in terms of dy and dx.

Step 2: Separate the Variables

Rewrite the equation to separate all terms involving \( y \) on one side and all terms involving \( x \) on the other side. Start by writing:\[ \frac{dy}{dx} = \frac{y}{1+y^2} \]Next, rearrange to separate variables:\[ \frac{1+y^2}{y} \, dy = dx \]

Step 3: Integrate Both Sides

Integrate both sides of the equation:\[ \int \frac{1+y^2}{y} \, dy = \int \, dx \]The left side separates into two integrals:\[ \int \left( \frac{1}{y} + y \right) dy = \ln|y| + \frac{y^2}{2} \]The right side integrates to:\[ x + C \] (where \( C \) is the integration constant).

Step 4: Solve for the General Solution

Combine the results from the integration to find the general solution:\[ \ln|y| + \frac{y^2}{2} = x + C \]This cannot be simplified to express \( y \) explicitly.

Step 5: Use CAS to Plot Integral Curves

A Computer Algebra System (CAS) can implicitly plot the integral curves for different values of \( C \). You can input the equation \( \ln|y| + \frac{y^2}{2} = x + C \) into a CAS tool and generate plots by selecting various values for \( C \). Choose five different constant values for \( C \) to plot five integral curves.

Key Concepts

Separable Differential Equations

A separable differential equation is a type of equation where you can separate the variables, enabling the integration of each side independently. This format is often seen in the form \( \frac{dy}{dx} = g(y)h(x) \), where variables \( y \) and \( x \) can be placed on opposite sides of the equation.
In the exercise, the differential equation given is \( y' = \frac{y}{1+y^2} \). Recognizing it as separable means you can rearrange the terms to:
\[ \frac{1+y^2}{y} \, dy = dx \]
This step facilitates the process of finding solutions by integrating each side separately. After rearranging, you can integrate the terms involving \( y \) over \( dy \) and the terms involving \( x \) over \( dx \).
Through integration, you get:
\[ \ln|y| + \frac{y^2}{2} = x + C \]
where \( C \) is the integration constant. Unlike some other differential equations, you might not always find an explicit solution for \( y \). But, finding a relation like the equation above is a valid and important solution step.

Integral Curves

Integral curves are graphs of solutions to a differential equation for various initial conditions. They represent the trajectory or path traced in the xy-plane by a function satisfying the differential equation.
In this problem, after getting the implicit solution \( \ln|y| + \frac{y^2}{2} = x + C \), each value of \( C \) gives a different curve. These curves interrelate to show how the chosen initial conditions affect the pathways or solutions that the differential equation models.
Integral curves visualized can provide deep insights into the behavior of the solutions, showing patterns and symmetry that might not be evident from just the equation itself. These insights are critical when analyzing dynamic systems, helping students and researchers predict behavior under various scenarios.

Computer Algebra System (CAS)

A Computer Algebra System (CAS) is an invaluable tool in solving and graphing complex mathematical problems, including differential equations. CAS software assists in performing symbolic calculations, simplifying expressions, and plotting curves which could be time-consuming or difficult by hand.
In this exercise, a CAS can be used to plot integral curves of the differential equation \( \ln|y| + \frac{y^2}{2} = x + C \) by selecting different constants \( C \).

• Through implicit plotting, CAS allows for a comprehensive understanding of the solutions.
• It handles algebraic manipulations smoothly and presents multiple integral curves efficiently.
• This visualization aids in verifying the general solution's behavior across different conditions.

Utilizing a CAS not only saves time but also enhances understanding, making it a powerful resource for students tackling intricate differential equations.