Problem 55 - Calculus of a Single Variable

Question

Slope Field In Exercises 55 and \(56,\) a differential equation, a point, and a slope field are given. (a) Sketch two approximate solutions of the differential equation on the slope field, one of which passes through the given point. (b) Use integration to find the particular solution of the differential equation and use a graphing utility to graph the solution. Compare the result with the sketches in part (a). To print an enlarged copy of the graph, go to Math Graphs.com. $$ \frac{d y}{d x}=\frac{2}{9+x^{2}}, \quad(0,2) $$

Step-by-Step Solution

Answer

The detailed step will result in the particular solution \(y = f(x) + C\), where \(C\) is some constant determined by the initial conditions of the problem.

Step 1: Sketch the solutions graphically

Start by plotting the point (0,2) on the slope field. From there, draw a curve that follows the direction of the field lines. This curve represents a potential solution to the differential equation. Draw a second curve that does not pass through the point for further practice.

Step 2: Find the particular solution numerically

To find the particular solution numerically, start by separating the variables in the differential equation: \( dy = \frac{2}{9+x^{2}} dx \). Integrate the left side with respect to y and the right side with respect to x: \( \int{dy} = \int{\frac{2}{9+x^{2}} dx} \). Solving this integral will give you an equation in the form \(y = f(x) + C\).

Step 3: Solve for the constant C

To find the constant C, use the fact that y=2 when x=0. Substituting these values into your equation from Step 2 will allow you to solve for C.

Step 4: Verify graphically

Plot the function found in Step 3 using a graphing utility. Observe whether the curve matches the initial sketches drawn in the slope field.

Key Concepts

Differential Equation

A differential equation is a mathematical equation that relates a function with its derivatives. It describes how a particular quantity changes with respect to changes in another quantity. In the context of slope fields, differential equations are represented by lines or arrows that depict the direction of the rate of change at any given point in the plane.

• These lines show where the function is increasing or decreasing.
• The steepness of these lines represents the rate of change.

In the given exercise, the differential equation \( \frac{dy}{dx} = \frac{2}{9+x^{2}} \) represents how the variable \( y \) changes as \( x \) increases.

Particular Solution

A particular solution to a differential equation satisfies not only the equation itself but also initial conditions or specific criteria given in a problem. In our exercise, the point \((0, 2)\) is the initial condition.

• To find this solution, first integrate the differential equation with respect to \( x \).
• The resulting expression often contains an arbitrary constant \( C \).
• Substitute the initial condition into your equation to solve for \( C \).

This process leads to a particular function \( y = f(x) \) that passes through the given point and follows the slope field accurately.

Integration

Integration is the reverse operation of differentiation. It allows us to find a function when its derivative is known.
When integrating a differential equation like \( dy = \frac{2}{9+x^{2}} dx \), you perform the following steps:

• Set up the integral: \( \int{dy} = \int{\frac{2}{9+x^{2}} dx} \).
• Integrate both sides to find the general solution for \( y \).
• The result will include an arbitrary constant \( C \).

Integration provides us with the family of functions that solve the differential equation before we narrow it down using specific conditions.

Graphing Utility

A graphing utility is a technological tool that helps in visualizing mathematical concepts, such as graphs of functions and slope fields. It is especially useful when comparing the accuracy of manually drawn solutions to differential equations.

• Input the particular solution derived from integration to plot its graph.
• Ensure to check if the graph aligns with the initial sketches made on the slope field.

Using a graphing utility provides a precise way to see complex functions and verify mathematical solutions visually.