/*! 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 5 State the order of the different... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 10: Problem 5

State the order of the differential equation and verify that the given function is a solution. $$\left(1-t^{2}\right) y^{\prime \prime}-2 t y^{\prime}+2 y=0, y(t)=t$$

Short Answer

Expert verified
The order is 2. The function \( y(t) = t \) is a solution.

Step by step solution

01

Identify the Differential Equation

Given the differential equation: \[ \left(1-t^{2}\right) y^{\prime \prime}-2 t y^{\prime}+2 y=0 \]
02

State the Order of the Differential Equation

The order of a differential equation is determined by the highest order derivative in the equation. In this case, the highest order derivative is \( y^{\prime \prime} \), which is the second derivative. Therefore, the differential equation is of order 2.
03

Verify the Function is a Solution

The proposed solution is \( y(t)=t \). First, find the first and second derivatives of \( y(t) \): \[ y^{\prime}(t) = 1, \quad y^{\prime \prime}(t) = 0 \] Substitute these derivatives into the differential equation: \[ (1-t^2) \cdot 0 - 2t \cdot 1 + 2t = 0 \] Simplify the expression: \[ 0 - 2t + 2t = 0 \] This verifies that \( y(t) = t \) satisfies the differential equation.

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 Equation
A differential equation is an equation that involves an unknown function and its derivatives. These equations are powerful tools in mathematics and are used to model real-life scenarios such as physics phenomena, engineering problems, and more. In the context of our exercise, the differential equation given is:
\( \left(1-t^{2}\right) y^{\prime \prime}-2 t y^{\prime}+2 y=0 \)
This is a second-order differential equation, meaning that the highest derivative is the second derivative (\

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

A certain drug is administered intravenously to a patient at the continuous rate of \(r\) milligrams per hour. The paticnt's body removes the drug from the bloodstream at a rate proportional to the amount of the drug in the blood, with constant of proportionality \(k=.5\) (a) Write a differential equation that is satisfied by the amount \(f(t)\) of the drug in the blood at time \(t\) (in hours). (b) Find \(f(t)\) assuming that \(f(0)=0 .\) (Give your answer in terms of \(r .)\) (c) In a therapeutic 2 -hour infusion, the amount of drug in the body should reach 1 milligram within 1 hour of administration and stay above this level for another hour. However, to avoid toxicity, the amount of drug in the body should not exceed 2 milligrams at any time. Plot the graph of \(f(t)\) on the interval \(1 \leq t \leq 2,\) as \(r\) varies between 1 and 2 by increments of \(.1 .\) That is, plot \(f(t)\) for \(r=1,1.1,1.2,1.3, \ldots . .2 .\) By looking at the graphs, pick the values of \(r\) that yield a therapeutic and nontoxic 2-hour infusion.

Solve the following differential equations with the given initial conditions. $$\frac{d y}{d t}=\left(\frac{1+t}{1+y}\right)^{2}, y(0)=2$$

Solving the differential equations that arise from modeling may require using integration by parts. [See formula (1).] After depositing an initial amount of \(\$ 10,000\) in a savings account that earns \(4 \%\) interest compounded continuously, a person continued to make deposits for a certain period of time and then started to make withdrawals from the account. The annual rate of deposits was given by \(3000-500 t\) dollars per year, \(t\) years from the time the account was opened. (Here, negative rates of deposits correspond to withdrawals.) (a) How many years did the person contribute to the account before starting to withdraw money from it? (b) Let \(P(t)\) denote the amount of money in the account, \(t\) years after the initial deposit. Find an initial-value problem satisfied by \(P(t)\). (Assume that the deposits and withdrawals were made continuously.)

Solve the given equation using an integrating factor. Take \(t>0\). $$e^{t} y^{\prime}+y=1$$

Suppose that \(f(t)\) is a solution of the differential equation \(y^{\prime}=t y-5\) and the graph of \(f(t)\) passes through the point (2,4). What is the slope of the graph at this point?
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Calculus

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Mechanics Maths

Read Explanation

Statistics

Read Explanation

Decision Maths

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.