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Chapter 1: Problem 16

Determine by inspection at least one solution for the given differential equation. $$ y^{*}=y^{3}-8 $$

Short Answer

Expert verified
The solution by inspection is \( y = 2 \).

Step by step solution

01

Understand the Differential Equation

The given equation is a first-order differential equation: \( y' = y^3 - 8 \). Here, \( y' \) represents the derivative of \( y \) with respect to another variable, often \( x \).
02

Consider Constant Solutions

To find solutions by inspection, let's first consider constant solutions where \( y' = 0 \). In such cases, the equation simplifies to \( y^3 - 8 = 0 \). We need to solve \( y^3 = 8 \) to find any constant solutions.
03

Solve for Constant \( y \)

Solve the equation \( y^3 = 8 \). By taking the cube root of both sides, we find: \( y = 2 \). Thus, \( y = 2 \) is a constant solution, which satisfies the differential equation by making \( y' = 0 \).
04

Verify the Solution

Verify that \( y = 2 \) is a solution of the differential equation. Substitute \( y = 2 \) into the original equation: \( y' = 2^3 - 8 = 8 - 8 = 0 \). Since \( y' = 0 \), the solution satisfies the equation, confirming it as a valid solution.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

First-Order Differential Equations
First-order differential equations are mathematical expressions that relate a function and its first derivative. These equations, denoted as \( y' = f(x, y) \), describe how a function \( y \) changes in relation to a variable \( x \). Such equations are powerful in modeling a variety of real-world scenarios, from physics to economics. In the given exercise, the differential equation \( y' = y^3 - 8 \) is a typical example of a first-order differential equation. Here, \( y' \) represents the rate of change of \( y \) with respect to \( x \). Understanding the structure of these equations is crucial for finding their solutions.
Constant Solutions
Constant solutions are special solutions of a differential equation where the derivative \( y' \) equals zero. This implies that the quantity \( y \) remains unchanged鈥攊t is a constant. To find a constant solution of the equation \( y' = y^3 - 8 \), we set \( y^3 - 8 = 0 \). This simplifies to \( y^3 = 8 \), thus by taking the cube root, we solve for \( y = 2 \).
Constant solutions are significant because they often represent equilibrium states or steady states in a system, where despite dynamic forces at play, the state remains unchanged over time.
Derivative Analysis
Derivative analysis is a valuable tool when examining differential equations. It involves exploring the relationship between a function and its rate of change. In our differential equation, \( y' = y^3 - 8 \), derivative analysis requires determining how \( y' \) behaves for different values of \( y \).
By inspecting where \( y' = 0 \), we identify potential constant solutions since the function exhibits no change at these points. In this exercise, setting \( y' = 0 \) led us to find \( y = 2 \) as a constant solution, by solving \( y^3 - 8 = 0 \). Derivative analysis thus helps us understand the behavior of the solutions and their dependencies on the variables involved.
Solution Verification
Solution verification is the essential step of confirming whether a proposed solution truly satisfies the differential equation. For the solution verification process, we substitute the constant solution \( y = 2 \) back into the original equation \( y' = y^3 - 8 \).
By doing this substitution, we obtain \( y' = 2^3 - 8 = 8 - 8 = 0 \). This satisfies the requirement that \( y' = 0 \) exactly, confirming \( y = 2 \) as a valid solution. Verification ensures that potential solutions indeed fulfill the equation, thereby empowering us to trust the reliability of our mathematical results.

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Most popular questions from this chapter

Classify the given differential equation as to type and order. Classify the ordinary differential equations as to linearity. $$ x^{2} \frac{d^{2} y}{d x^{2}}-3 x \frac{d y}{d x}+y=x^{2} $$

Determine by inspection at least two solutions of the given initial-value problem. $$ x \frac{d y}{d x}=2 y, \quad y(0)=0 $$

\(y^{\prime}+2 x y=1 ; \quad y=e^{-x^{2}} \int_{0}^{x} e^{t^{2}} d t+c_{1} e^{-x^{2}}\)

Suppose that a large mixing tank initially holds 300 gallons of water in which 50 pounds of salt has been dissolved. Pure water is pumped into the tank at a rate of \(3 \mathrm{gal} / \mathrm{min}\), and then when the solution is well stirred it is pumped out at the same rate. Determine a differential equation for the amount \(A(t)\) of salt in the tank at any time \(t\).

A large snowball is shaped into the form of a sphere. Starting at some time, which we can designate as \(t=0\), the snowball begins to melt. Assume for the sake of discussion that the snowball melts in such a manner that its shape remains spherical. Discuss the quantities that change with time as the snowball melts. Discuss an interpretation of "melting" as a rate. If possible, construct a mathematical model that describes the state of the snowball at any time \(t>0\).
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