Problem 60 - Calculus of a Single Variable

Question

Slope Field In Exercises \(57-60\) , use a computer algebra system to graph the slope field for the differential equation and graph the solution satisfying the specified initial condition. $$ \begin{array}{l}{\frac{d y}{d x}=\frac{\sqrt{y}}{1+x^{2}}} \\\ {y(0)=4}\end{array} $$

Step-by-Step Solution

Answer

The short answer is the graphical representation of both the slope field for the differential equation \(\frac{dy}{dx} = \frac{\sqrt{y}}{1+x^{2}}\) and the solution curve that passes through the point \((0, 4)\). This solution can only be determined using the mentioned computer algebra systems, thus a specific graphical display cannot be provided in this textual format.

Step 1: Set Up the Differential Equation

The differential equation given in this problem is \(\frac{dy}{dx} = \frac{\sqrt{y}}{1+x^{2}}\). This equation gives the rate of change of \(y\) (the function to be discovered) at any point \(x\). The goal is to plot the slope field, i.e., the set of all possible slopes given by this differential equation.

Step 2: Solve for the Slope Field

A slope field is obtained by plotting the slope of the solution at different points in the \(x-y\) space. These slopes are determined by the differential equation. For a point \((x, y)\) in the plane, the slope of the function at \((x, y)\) is given by \(\frac{\sqrt{y}}{1+x^{2}}\). As this exercise suggests employing a computer algebra system, use software like MATLAB or Mathematica to enter the differential equation and generate the slope field.

Step 3: Solve and Graph the Solution with the Given Initial Condition

Use the computer algebra system to find the solution of the differential equation that satisfies the initial condition \(y(0) = 4\). This solution is a particular curve in the \(x-y\) plane that passes through the initial condition and whose slope at any point matches the corresponding point in the slope field. After solving the equation, plot this solution over the slope field.

Key Concepts

Differential Equations

Differential equations are mathematical equations that involve functions and their derivatives. They are vital in modeling real-world systems where quantities change with respect to each other. In our exercise, the differential equation is \( \frac{dy}{dx} = \frac{\sqrt{y}}{1+x^{2}} \). This gives the relationship between the rates of change of \( y \) with respect to \( x \).
Here's how it breaks down:

• The left side \( \frac{dy}{dx} \) denotes the derivative of \( y \) concerning \( x \), reflecting how \( y \) changes as \( x \) changes.
• The right side \( \frac{\sqrt{y}}{1+x^{2}} \) represents how this rate of change is dependent on both \( y \) and \( x \).

Such equations are used to predict and analyze the behavior of complex systems in physics, biology, economics, and more.

Initial Condition

An initial condition helps in finding a specific solution from a family of solutions to a differential equation. It is a given point that the solution curve must pass through.
In our exercise, the initial condition is \( y(0) = 4 \). This states that when \( x = 0 \), \( y \) must be 4.

• This unique point guides the particular solution in the slope field.
• Without it, we could have infinitely many solutions making it hard to pinpoint one specific curve.

Using initial conditions ensures that we can describe a precise scenario or outcome in modeling situations.

Computer Algebra System

A Computer Algebra System (CAS) is software designed to perform symbolic mathematics. In our exercise, a CAS is used to graph the slope field for the differential equation and to find a specific solution that satisfies the initial condition.

• Software like MATLAB or Mathematica can solve complicated equations swiftly and accurately.
• They help in visualizing the slope field, which plots tiny slope lines at various points in the \(x-y\) plane.

This visualization aids in understanding how solutions behave near different regions, making it a vital tool in learning and applying differential equations.

Rate of Change

Rate of change is a crucial concept in calculus and differential equations, representing how a variable changes with respect to another. In our exercise, the rate of change of \( y \) concerning \( x \) is defined by \( \frac{dy}{dx} = \frac{\sqrt{y}}{1+x^{2}} \).

• This expression tells us how fast \( y \) changes relative to \( x \).
• By interpreting these rates, we can predict the growing or shrinking behavior of \( y \) at different \( x \) values.

Understanding the rate of change helps in forecasting trends and making informed decisions in fields like engineering, science, and finance.