Problem 8 - Calculus: Early Transcendentals

Question

(a) What can you say about the graph of a solution of the equation \( y^{'} = xy^3 \) when \( x \) is close to \( 0? \) What if \( x \) is large? (b) Verify that all members of the family \( y = (c - x^2) ^{{-}{1/2}} \) are solutions of the differential equation \( y^{'} = xy^3. \) (c) Graph several members of the family of solutions on a common screen. Do the graphs confirm what you predicted in part (a)? (d) Find a solution of the initial-value problem \( y^{'} = xy^3 \) \( y(0) = 2 \)

Step-by-Step Solution

Answer

Near \( x = 0 \), the graph is flat; for large \( x \), it changes rapidly. The solution to the initial-value problem is \( y = (1/4 - x^2)^{-1/2} \).

Step 1: Analyzing the Behavior of the Graph as x Approaches 0

When considering the differential equation \( y^{'} = xy^3 \), as \( x \) approaches 0, \( y^{'} \) also tends to 0 regardless of the value of \( y \). This indicates that near \( x = 0 \), the slope of the solution curves become very small. Consequently, the graph of the solution will tend to be horizontal or nearly flat, as the change in \( y \) with respect to \( x \) becomes negligible.

Step 2: Analyzing the Behavior of the Graph as x Becomes Large

For large values of \( x \), the term \( xy^3 \) in the derivative indicates that the slope \( y^{'} \) is heavily dependent on both \( x \) and \( y \). If \( y \) remains bounded, then \( y^{'} \) grows with \( x \), resulting in solutions where \( y \) varies significantly with \( x \). Thus, the graph of the solution will exhibit steep or rapid changes for large \( x \).

Step 3: Verifying Family of Solutions

To verify that \( y = (c - x^2)^{-1/2} \) is a solution of \( y^{'} = xy^3 \), start by differentiating \( y \) with respect to \( x \):\[ y = (c - x^2)^{-1/2} \]Differentiating using the chain rule:\[ y^{'} = \frac{d}{dx}(c - x^2)^{-1/2} = \frac{1}{2}(c - x^2)^{-3/2}(-2x) = \frac{-x}{(c - x^2)^{3/2}} \]Now compute \( xy^3 \):\[ xy^3 = x\left((c - x^2)^{-1/2}\right)^3 = \frac{x}{(c - x^2)^{3/2}} \]Thus, \( y^{'} = xy^3 \), confirming that \( y = (c - x^2)^{-1/2} \) is a solution.

Step 4: Graphing Family of Solutions

By graphing several members of the family \( y = (c - x^2) ^{-1/2} \) for different constants \( c \), observe that near \( x = 0 \), the graphs are nearly flat as predicted. As \( x \) becomes large, the graphs demonstrate rapid changes in \( y \). This visual evidence supports the observations made in part (a) about the behavior of solutions.

Step 5: Solving the Initial Value Problem

To find a solution for the initial-value problem \( y^{'} = xy^3 \) with \( y(0) = 2 \), consider using the general solution family \( y = (c - x^2)^{-1/2} \).Substitute \( x = 0 \) and \( y(0) = 2 \):\[ y = (c - 0^2)^{-1/2} = (c)^{-1/2} = 2 \]Solving \( c^{-1/2} = 2 \) results in \( c = 1/4 \).Thus, the solution to the initial value problem is \( y = (1/4 - x^2)^{-1/2} \).

Key Concepts

Initial Value Problem

An initial value problem in differential equations consists of finding a specific solution that satisfies not only the differential equation itself but also an initial condition. This means you’re given a starting point, usually in the form of a specific value of the function at a certain point, such as \( y(0) = 2 \).
An initial value problem is vital because it narrows down to a unique solution among a family of solutions. In the provided exercise, you start with the differential equation \( y' = xy^3 \) and are tasked to satisfy the condition \( y(0) = 2 \).
To solve an initial value problem, pinpoint the family of solutions, like \( y = (c - x^2)^{-1/2} \) in this case, and apply the initial condition to determine the constant \( c \).

• Substitute \( x = 0 \) and \( y = 2 \) into the equation.
• Calculate to find \( c \).
• Use this \( c \) to get your particular solution.

This is how the particular solution \( y = (1/4 - x^2)^{-1/2} \) was discovered, ensuring it meets both the equation and the initial condition.

Graphing Solutions

Graphing solutions of differential equations allows us to visualize how solutions behave. This is especially helpful when analyzing the behavior over a range of values for \( x \).
For instance, in the exercise, solutions are graphed to observe behavior as \( x \) approaches 0 versus when it is large. Graphs can help you see patterns, such as:

• Near \( x = 0 \), the solutions \( y \) flatten, indicating a small slope or rate of change.
• For large \( x \) values, the graph shows steep slopes, suggesting rapid changes in \( y \).

By graphing multiple solutions with various constants \( c \), you can confirm the behavior predicted from the initial analysis in the exercise.
This visual method strengthens the understanding of theoretical predictions made about the solutions’ behavior across different values of \( x \).

Chain Rule

The chain rule is a fundamental calculus concept used to differentiate compositions of functions. It lets you find the derivative of functions nested within other functions.
In the differential equation given, the solution involves \( y = (c - x^2)^{-1/2} \). To differentiate \( y \) properly, the chain rule is applied.
Here’s how it works in this context:

• The function \((c - x^2)^{-1/2}\) is composed of \(f(g(x))\) where \( f(u) = u^{-1/2} \) and \( g(x) = c - x^2 \).
• Differentiating \( f(u) \) gives \( -1/2u^{-3/2} \).
• Differentiating \( g(x) \) yields \( -2x \).
• Combine them: \( y' = f'(g(x))g'(x) = \frac{-x}{(c-x^2)^{3/2}} \).

This confirms the original equation \( y' = xy^3 \) and shows how the solution fits as a valid derivative.

Solution Verification

Solution verification is the process of proving that a proposed solution actually satisfies the original differential equation. It’s an essential step to confirm your solution is not just mathematically correct but relevant to the problem.
In the exercise, we verify that \( y = (c - x^2)^{-1/2} \) satisfies the equation \( y' = xy^3 \). This involves calculating both sides of the equation independently:

• Compute \( y' \) using the derived expression \( \frac{-x}{(c - x^2)^{3/2}} \).
• Calculate \( xy^3 \), which simplifies to \( \frac{x}{(c - x^2)^{3/2}} \).

The expressions match, proving that the guessed function indeed solves the differential equation.
Verification is crucial because it backs up any equation manipulations or assumptions you might have made during the solving process, ensuring the solution's validity.