/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 8 Determine whether each function ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 11: Problem 8

Determine whether each function is a solution to the differential equation and justify your answer: $$x \frac{d y}{d x}=4 y$$ (a) \(y=x^{4}\) (b) \(y=x^{4}+3\) (c) \(y=x^{3}\) (d) \(y=7 x^{4}\)

Short Answer

Expert verified
Solutions: (a) \(y=x^4\), (d) \(y=7x^4\).

Step by step solution

01

Differentiate Each Function

(a) For \( y = x^4 \), the derivative is \( \frac{dy}{dx} = 4x^3 \).(b) For \( y = x^4 + 3 \), the derivative is \( \frac{dy}{dx} = 4x^3 \).(c) For \( y = x^3 \), the derivative is \( \frac{dy}{dx} = 3x^2 \).(d) For \( y = 7x^4 \), the derivative is \( \frac{dy}{dx} = 28x^3 \).
02

Substitute into Differential Equation

Substitute each function and its derivative back into the differential equation \( x \frac{dy}{dx} = 4y \).(a) Substitute \( \frac{dy}{dx} = 4x^3 \) and \( y = x^4 \): \[ x \cdot 4x^3 = 4(x^4) \quad \Rightarrow \quad 4x^4 = 4x^4 \] True.(b) Substitute \( \frac{dy}{dx} = 4x^3 \) and \( y = x^4 + 3 \): \[ x \cdot 4x^3 = 4(x^4 + 3) \quad \Rightarrow \quad 4x^4 = 4x^4 + 12 \] False.(c) Substitute \( \frac{dy}{dx} = 3x^2 \) and \( y = x^3 \): \[ x \cdot 3x^2 = 4(x^3) \quad \Rightarrow \quad 3x^3 = 4x^3 \] False.(d) Substitute \( \frac{dy}{dx} = 28x^3 \) and \( y = 7x^4 \): \[ x \cdot 28x^3 = 4(7x^4) \quad \Rightarrow \quad 28x^4 = 28x^4 \] True.
03

Conclude Which Functions Are Solutions

From the substitution, the functions \( y = x^4 \) and \( y = 7x^4 \) satisfy the differential equation, so they are solutions.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Derivatives
Understanding derivatives is crucial in solving differential equations. A derivative represents how a function changes as its input changes. In simpler terms, it captures the rate of change or the slope of the function at any point along its curve.
To determine if a function solves a differential equation, we first need to find its derivative. This involves using rules of differentiation and applying them to each specific function.
For instance:
  • In part (a), the derivative of the function \( y = x^4 \) is \( \frac{dy}{dx} = 4x^3 \).
  • In part (b), the derivative of \( y = x^4 + 3 \) remains \( 4x^3 \) since the derivative of a constant, like 3, is zero.
  • For \( y = x^3 \) in part (c), the derivative is \( 3x^2 \).
  • And for \( y = 7x^4 \) in part (d), the derivative becomes \( 28x^3 \) as a result of the constant multiplier.
Once these derivatives are known, they can be substituted into the differential equation to check for consistency.
Solution Verification
Verification of a solution to a differential equation involves substituting both the function and its derivative back into the equation to see if both sides are equal. This process ensures that no mathematical errors were made during differentiation.
For the given differential equation \( x \frac{dy}{dx} = 4y \), verification requires:
  • Substituting the derived \( \frac{dy}{dx} \) and the original function \( y \).
  • Confirming that the left-hand side (LHS) equals the right-hand side (RHS) of the equation.
For example:
  • In (a) where \( y = x^4 \), substitution results in \( 4x^4 = 4x^4 \), confirming it as a solution.
  • In (b) with \( y = x^4 + 3 \), LHS becomes \( 4x^4 \) while RHS is \( 4x^4 + 12 \), leading to a discrepancy.
  • In (c), the lack of equality \( 3x^3 eq 4x^3 \) clearly denotes it's not a solution.
  • For (d), perfect equality \( 28x^4 = 28x^4 \) reaffirms it as a correct solution.
Hence, verification shows which functions satisfy the differential equation.
Mathematical Justification
Justifying the solutions mathematically involves explaining why certain functions satisfy the differential equation, while others do not. A key aspect of this justification is examining any constants present in the functions.
  • Functions that maintain proportional relationships, like \( y = x^4 \) and \( y = 7x^4 \), are solutions because their derivatives maintain the equation's balance.
  • Functions with additive constants, such as \( y = x^4 + 3 \), disrupt the balance after differentiation, resulting in an inconsistency in values when substituted back.
  • Simplicity plays a role as well. \( y = x^3 \) fails because its derivative's factor does not align with the given differential equation demands.
  • Multiple solutions often indicate the differential equation's allowance for scalar multiples, as shown with \( y = 7x^4 \).
Justification gives insights into the structural requirements of valid solutions.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

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 \(\lim _{x \rightarrow \infty} g(x)=\infty,\) then \(\lim _{x \rightarrow \infty} f(x)=\infty.\)

A model for the population. \(P\), of carp in a landlocked lake at time \(t\) is given by the differential equation $$\frac{d P}{d t}=0.25 P(1-0.0004 P)$$ (a) What is the long-term equilibrium population of carp in the lake? (b) A census taken ten years ago found there were 1000 carp in the lake. Estimate the current population. (c) Under a plan to join the lake to a nearby river, the fish will be able to leave the lake. A net loss of \(10 \%\) of the carp each year is predicted, but the patterns of birth and death are not expected to change. Revise the differential equation to take this into account. Use the revised differential equation to predict the future development of the carp population.

As you know, when a course ends, students start to forget the material they have learned. One model called the Ebbinghaus model) assumes that the rate at which a student forgets material is proportional to the difference between the material currently remembered and some positive constant, \(a\) (a) Let \(y=f(t)\) be the fraction of the original material remembered \(t\) weeks after the course has ended. Set up a differential equation for \(y .\) Your equation will contain two constants; the constant \(a\) is less than \(y\) for all \(t.\) (b) Solve the differential equation. (c) Describe the practical meaning (in terms of the amount remembered) of the constants in the solution \(y=f(t).\)

Let \(L,\) a constant, be the number of people who would like to see a newly released movie, and let \(N(t)\) be the number of people who have seen it during the first \(t\) days since its release. The rate that people first go see the movie, \(d N / d t\) (in people/day), is proportional to the number of people who would like to see it but haven't yet. Write and solve a differential equation describing \(d N / d t\) where \(t\) is the number of days since the movie's release. Your solution will involve \(L\) and a constant of proportionality, \(k.\)

An aquarium pool has volume \(2 \cdot 10^{6}\) liters. The pool initially contains pure fresh water. At \(t=0\) minutes, water containing 10 grams/liter of salt is poured into the pool at a rate of 60 liters/minute. The salt water instantly mixes with the fresh water, and the excess mixture is drained out of the pool at the same rate (60 liters/minute). (a) Write a differential equation for \(S(t),\) the mass of salt in the pool at time \(t.\) (b) Solve the differential equation to find \(S(t).\) (c) What happens to \(S(t)\) as \(t \rightarrow \infty ?\)
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Applied Mathematics

Read Explanation

Geometry

Read Explanation

Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Calculus

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.