/*! 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} Q11RP The initial value problem \[\fra... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Q11RP (page 30)

The initial value problem \[\frac{{{\bf{dx}}}}{{{\bf{dt}}}}{\bf{ = 3}}{{\bf{x}}^{\frac{{\bf{2}}}{{\bf{3}}}}}{\bf{,}}\;{\bf{x(0) = 1}}\] has a unique solution in some open interval around t = 0.

Short Answer

Expert verified

The given statement is true.

Step by step solution

01

Finding partial derivatives

Since \[\frac{{{\bf{dx}}}}{{{\bf{dt}}}}{\bf{ = 3}}{{\bf{x}}^{\frac{{\bf{2}}}{{\bf{3}}}}}\]

Then\[\frac{{\partial {\bf{f}}}}{{\partial {\bf{x}}}}{\bf{ = }}\frac{{\bf{2}}}{{\sqrt[{\bf{3}}]{{\bf{x}}}}}\]

02

Checking the final result

Apply the initial conditions\[x\left( 0 \right) = 1\]

\[\frac{{\partial f}}{{\partial x}}{\bf{ = }}\frac{{\bf{2}}}{{\bf{0}}}\]

The result is infinite.

The given function is discontinuous a x = 0. So the function is not continuous in a rectangle containing the point (0,1).

Therefore, the statement is True.

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

In problems 1-6, identify the independent variable, dependent variable, and determine whether the equation is linear or nonlinear.

3r-cos胃drd胃=sin胃

In Problems 9-13, determine whether the given relation is an implicit solution to the given differential equation. Assume that the relationship implicitly defines y as a function of x and use implicit differentiation.

y-logey=x2+1,dydx=2xyy-1

In Problems 9-13, determine whether the given relation is an implicit solution to the given differential equation. Assume that the relationship implicitly defines y as a function of x and use implicit differentiation.

siny+xy-x3=2,y''=6xy'+y'3siny-2y'23x2-y

Question: Let f(x)and g(x)be analytic at x0. Determine whether the following statements are always true or sometimes false:

(a) 3f(x)+g(x)is analytic at x0.

(b) f(x)/g(x)is analytic at x0.

(c) f'(x)is analytic at x0.

(d) f(x)3-鈭�x0xg(t)dt is analytic at x0.

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).
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Discrete Mathematics

Read Explanation

Applied Mathematics

Read Explanation

Pure Maths

Read Explanation

Theoretical and Mathematical Physics

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.