/*! 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} Q23 E In Problems 23-28, determine whe... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Q23 E (page 1)

In Problems 23-28, determine whether Theorem 1 implies that the given initial value problem has a unique solution.

dydx=y4-x4,y(0)=7

Short Answer

Expert verified

The hypotheses of Theorem 1 are satisfied.

The theorem shows that the given initial value problem has a unique solution.

Step by step solution

01

Finding the partial derivative of the given relation with respect to y

Here, fx,y=y4-x4and鈭�f鈭�y=4y3.

02

Determining whether Theorem 1 implies the existence of a unique solution or not.

Now from Step 1, we find that both of the functionsfx,yand鈭�f鈭�yare continuous in any rectangle containing the point0,7, so the hypotheses of the Theorem are satisfied. It then follows from the theorem that the given initial value problem has a unique solution in an interval aboutx=0of the form0-未,0+未, where未is some positive number.

Hence, Theorem 1 implies that the given initial value problem has a unique solution.

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!

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

Implicit Function Theorem. Let G(x,y)have continuous first partial derivatives in the rectangleR={x,y:a<x<b,c<y<d}containing the pointlocalid="1664009358887" (x0,y0). IfG(x0,y0)=0 and the partial derivativeGy(x0,y0)鈮�0, then there exists a differentiable function y=蠒(x), defined in some intervalI=(x0-未,y0+未),that satisfies G for allforG(x,蠒x)all x鈭�I.

The implicit function theorem gives conditions under which the relationshipG(x,y)=0 implicitly defines yas a function of x. Use the implicit function theorem to show that the relationshipx+y+exy=0 given in Example 4, defines y implicitly as a function of x near the point(0,-1).

Decide whether the statement made is True or False. The relation x2+y3-ey=1 is an implicit solution to dydx=ey-2x3y2.

Question:Use a CAS to graphJ3/2(x),J-3/2(x),J5/2(x), and J-5/2(x).

Consider the differential equation dydx=x+siny

猞� A solution curve passes through the point (1,蟺2). What is its slope at this point?

猞� Argue that every solution curve is increasing for x>1.

猞� Show that the second derivative of every solution satisfies d2ydx2=1+xcosy+12sin2y.

猞� A solution curve passes through (0,0). Prove that this curve has a relative minimum at (0,0).

Oscillations and Nonlinear Equations. For the initial value problem x''+(0.1)(1-x2)x'+x=0;x(0)=xo,x'(0)=0using the vectorized Runge鈥揔utta algorithm with h = 0.02 to illustrate that as t increases from 0 to 20, the solution x exhibits damped oscillations when xo=1, whereas exhibits expanding oscillations when xo=2.1,.
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Applied Mathematics

Read Explanation

Pure Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Calculus

Read Explanation

Probability and Statistics

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.