/*! 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 1 Find an integrating factor for e... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 10: Problem 1

Find an integrating factor for each equation. Take \(t>0\). $$ y^{\prime}-2 y=t $$

Short Answer

Expert verified
The integrating factor is \(e^{-2t}\).

Step by step solution

01

Identify the standard form

First, rewrite the given differential equation in the standard linear form. A standard first-order linear differential equation is written as: \[y^{\'}} + P(t)y = Q(t)\]For the given equation, compare and identify \(P(t)\) and \(Q(t)\):\[y^{\'}} - 2y = t\]Here, \(P(t) = -2\) and \(Q(t) = t\).
02

Determine the integrating factor \(\mu(t)\)

The integrating factor \(\mu(t)\) is found using the formula:\[\mu(t) = e^{\int P(t) \, dt}\]Given \(P(t) = -2\), we find:\[\int P(t) \, dt = \int -2 \, dt = -2t\]Therefore, the integrating factor is:\[\mu(t) = e^{-2t}\]
03

Verify the integrating factor calculation

To ensure correctness, verify the calculation of \(\mu(t)\):\[\mu(t) = e^{-2t}\]This means when multiplied through the original differential equation, it will facilitate solving it more easily by making the left-hand side an exact derivative.

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.

Differential Equations
Differential equations are mathematical equations that involve functions and their derivatives. They describe how quantities change and are fundamental in understanding many fields such as physics, engineering, economics, and biology.

A differential equation can relate the change in a quantity (described by the derivative) to the quantity itself. For example, the equation provided: \[ y^{\boldsymbol{\text{'}}} - 2y = t \] links the derivative of y with y itself and the variable t.

Differential equations can be classified by their order, linearity, and the nature of their solutions. Learning how to solve them is crucial for understanding complex systems and predicting their behavior.
Linear Differential Equation
A linear differential equation is a type of differential equation where the unknown function and its derivatives appear linearly. This means that the function and its derivatives are to the power of one and not multiplied together.

In our example, the given equation is: \[ y^{\boldsymbol{\text{'}}} - 2y = t \] This is a linear differential equation because y and its derivative appear to the first power and are not multiplied by each other.

The general form of a first-order linear differential equation is: \[ y^{\boldsymbol{\text{'}}} + P(t)y = Q(t) \] Here, P(t) and Q(t) are functions of t. The solution method involves finding an integrating factor, which simplifies the equation into something easier to integrate.
First-Order Differential Equations
First-order differential equations involve the first derivative of the unknown function but no higher derivatives. These equations are simpler to solve than higher-order ones.

For example, \[ y^{\boldsymbol{\text{'}}} - 2y = t \] is a first-order differential equation because it includes only the first derivative of y.

The process to solve this involves several steps:
  • Identifying the standard form.
  • Finding the integrating factor.
  • Solving the simplified equation.
For the given equation, we found the integrating factor to be: \[ \mu(t) = e^{-2t} \] This simplifies the problem, making it easier to integrate and find the solution.

Understanding how to handle first-order differential equations is essential for laying the foundation for more complex problems.

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

You are given a logistic equation with one or more initial conditions. (a) Determine the carrying capacity and intrinsic rate. (b) Sketch the graph of \(\frac{d N}{d t}\) versus \(N\) in an \(N z\) -plane. (c) In the \(t N\) -plane, plot the constant solutions and place a dashed line where the concavity of certain solutions may change. (d) Sketch the solution curve corresponding to each given initial condition. $$ d N / d t=-.01 N^{2}+N, N(0)=5 $$

You are given a logistic equation with one or more initial conditions. (a) Determine the carrying capacity and intrinsic rate. (b) Sketch the graph of \(\frac{d N}{d t}\) versus \(N\) in an \(N z\) -plane. (c) In the \(t N\) -plane, plot the constant solutions and place a dashed line where the concavity of certain solutions may change. (d) Sketch the solution curve corresponding to each given initial condition. $$ d N / d t=-N^{2}+N, N(0)=.5 $$

Solve the initial-value problem. $$ y^{\prime}+2 y=1, y(0)=1 $$

A person took out a loan of $$\$ 100,000$$ from a bank that charges \(7.5 \%\) interest compounded continuously. What should be the annual rate of payments if the loan is to be paid in full in exactly 10 years? (Assume that the payments are made continuously throughout the year.)

The Federal Housing Finance Board reported that the national average price of a new one-family house in 2012 was $$\$ 278,900$$. At the same time, the average interest rate on a conventional 30 -year fixedrate mortgage was \(3.1 \% .\) A person purchased a home at the average price, paid a down payment equal to \(10 \%\) of the purchase price, and financed the remaining balance with a 30 -year fixed-rate mortgage. Assume that the person makes payments continuously at a constant annual rate \(A\) and that interest is compounded continuously at the rate of \(3.1 \% .\) (Source: The Federal Housing Finance Board, www.fhfb.gov.) (a) Set up a differential equation that is satisfied by the amount \(f(t)\) of money owed on the mortgage at time \(t\) (b) Determine \(A\), the rate of annual payments, that is required to pay off the loan in 30 years. What will the monthly payments be? (c) Determine the total interest paid during the 30 -year term mortgage.
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Decision Maths

Read Explanation

Calculus

Read Explanation

Statistics

Read Explanation

Pure Maths

Read Explanation

Logic and Functions

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.