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Chapter 11: Problem 1

Is \(y=x^{3}\) a solution to the differential equation $$x y^{\prime}-3 y=0 ?$$

Short Answer

Expert verified
Yes, \( y = x^3 \) is a solution to the differential equation.

Step by step solution

01

Differentiate the given function

Start by finding the derivative of the function \( y = x^3 \). The derivative with respect to \( x \) is \( y' = \frac{d}{dx}(x^3) = 3x^2 \).
02

Substitute into the differential equation

Substitute \( y = x^3 \) and \( y' = 3x^2 \) into the differential equation \( x y' - 3y = 0 \). This gives \( x(3x^2) - 3(x^3) = 0 \).
03

Simplify the equation

Simplify the equation from Step 2: \( 3x^3 - 3x^3 = 0 \).
04

Verify the solution

Since \( 3x^3 - 3x^3 = 0 \) simplifies to \( 0 = 0 \), which is true, \( y = x^3 \) is indeed a solution to the differential equation.

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Key Concepts

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

Solutions to Differential Equations
Differential equations are mathematical equations that involve functions and their derivatives. They describe how a particular quantity changes with respect to another. Understanding solutions to these equations is crucial in modeling real-world phenomena like heat transfer, population dynamics, and more.

For a function to solve a differential equation, it must satisfy the equation when substituted into it. This implies that if you replace the function and its derivatives into the differential equation, you end up with a true statement.
  • Substitute the given function and its derivative into the original equation.
  • Simplify the resulting equation.
  • If the left side of the equation equals the right side, the function is indeed a solution.

This approach helps in verifying whether a proposed function behaves as expected according to the differential equation model.
Derivative Calculation
A derivative represents the rate at which a function changes. Calculating derivatives is a fundamental concept in understanding how functions behave, particularly in differentiating curves and finding tangent lines.

For a given function like \( y = x^3 \), the derivative \( y' \) indicates how \( y \) changes with respect to \( x \). Here is how you differentiate \( y = x^3 \):
  • Know the differentiation rule: the power rule is useful here, which states \( \frac{d}{dx}(x^n) = nx^{n-1} \).
  • Apply it to \( y = x^3 \), resulting in \( y' = 3x^2 \).

The derivative \( y' = 3x^2 \) is crucial for substitution into the differential equation to test if the function \( y \) satisfies it.
Function Verification
After calculating the derivative, the next step is function verification. You need to confirm whether your function truly solves the differential equation. Verification involves a logical process:

Start with substitution:
  • Insert both \( y = x^3 \) and its derivative \( y' = 3x^2 \) into the differential equation \( xy' - 3y = 0 \).
  • You get the equation \( x(3x^2) - 3(x^3) = 0 \).
  • Simplify to find \( 3x^3 - 3x^3 = 0 \), which further simplifies to \( 0 = 0 \).

Since the equation holds true, this confirms the function \( y = x^3 \) is indeed a solution. Understanding this process ensures you have correctly identified solutions to differential equations.

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Most popular questions from this chapter

Give the solution to the logistic differential equation with initial condition. $$\frac{d P}{d t}=0.8 P\left(1-\frac{P}{8500}\right) \text { with } P_{0}=500$$

Policy makers are interested in modeling the spread of information through a population. For example, agricultural ministries use models to understand the spread of technical innovations or new seed types through their countries. Two models, based on how the information is spread, follow. Assume the population is of a constant size \(M\) (a) If the information is spread by mass media (TV, radio, newspapers), the rate at which information is spread is believed to be proportional to the number of people not having the information at that time. Write a differential equation for the number of people having the information by time \(t .\) Sketch a solution assuming that no one (except the mass media) has the information initially. (b) If the information is spread by word of mouth, the rate of spread of information is believed to be proportional to the product of the number of people who know and the number who don't. Write a differential equation for the number of people having the information by time \(t .\) Sketch the solution for the cases in which (i) No one \(\quad\) (ii) \(5 \%\) of the population (iii) \(75 \%\) of the population knows initially. In each case, when is the information spreading fastest?

(a) Sketch the slope field for \(y^{\prime}=x / y\) (b) Sketch several solution curves. (c) Solve the differential equation analytically.

Is the statement true or false? Assume that \(y=f(x)\) is a solution to the equation \(d y / d x=g(x) .\) If the statement is true, explain how you know. If the statement is false, give a counterexample. If \(g(x)\) is periodic, then \(f(x)\) is also periodic.

Decide whether the statement is true or false. Assume that \(y=f(x)\) is a solution to the equation \(d y / d x=2 x-y .\) Justify your answer. $$f^{\prime}(x)=2 x-f(x)$$
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