(a) Explanation Procedure used: Existence of a Unique Solution: Let R be a rectangular region in the plane- x y defined by a ≤ x ≤ b , c ≤ y ≤ d , that contains the point ( x 0 , y 0 ) in its interior. If f ( x , y ) and δ f δ y are continuous on R, then there exists some interval I : ( x 0 − h , x 0 + h ) , h > 0 , contained in [ a , b ] , and a unique function y ( x ) defined in I that is a solution of the initial-value problem. The solution of the differential equation d y d x + P ( x ) y = f ( x ) can be expressed in the form of y ⋅ I . F . = ∫ f ( x ) ⋅ I . F . ⋅ d x + c where, I . F . = e ∫ P d x . Calculation: Consider the given equation. x y ′ − 4 y = x 6 e x y ′ − 4 x y = x 5 e x (1) The solution of the differential equation can be given as. y ⋅ I . F . = ∫ f ( x ) ⋅ I . F . ⋅ d x + c (2) Substitute e ∫ P d x for I . F . in equation (2) (b) To determine Explain the existence and uniqueness of a solution of IVP consisting of x y ′ − 4 y = x 6 e x for the given initial condition y ( 0 ) = y 0 , y 0 > 0 . (c) To determine Explain the existence and uniqueness of a solution of IVP consisting of x y ′ − 4 y = x 6 e x for the given initial condition y ( x 0 ) = y 0 , y 0 > 0 .

Differential Equations with Boundary-Value Problems (MindTap Course List)

9th Edition

ISBN: 9781305965799

Author: Dennis G. Zill

Publisher: Cengage Learning

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Differential Equations

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Chapter 2.3, Problem 50E

(a)

To determine

Explain the existence and uniqueness of a solution of IVP consisting of xy′−4y=x6ex for the given initial condition y(0)=0.

(b)

To determine

Explain the existence and uniqueness of a solution of IVP consisting of xy′−4y=x6ex for the given initial condition y(0)=y0,y0>0.

(c)

To determine

Explain the existence and uniqueness of a solution of IVP consisting of xy′−4y=x6ex for the given initial condition y(x0)=y0,y0>0.

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Chapter 2.3, Problem 49E

Chapter 2.3, Problem 51E

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Chapter 2 Solutions

Differential Equations with Boundary-Value Problems (MindTap Course List)

Ch. 2.1 - In Problems 14 reproduce the given...Ch. 2.1 - In Problems 14 reproduce the given...Ch. 2.1 - dydx=1xy (a) y(0) = 0 (b) y(1) = 0 (c) y(2) = 2...Ch. 2.1 - In Problems 14 reproduce the given...Ch. 2.1 - Prob. 5E Ch. 2.1 - Prob. 6E Ch. 2.1 - Prob. 7E Ch. 2.1 - Prob. 8E Ch. 2.1 - Prob. 9E Ch. 2.1 - Prob. 10E

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Explain the existence and uniqueness of a solution of IVP consisting of x y ′ − 4 y = x 6 e x for the given initial condition y ( 0 ) = 0 .

Solution Summary: The author explains the existence and uniqueness of a solution of IVP consisting of xyprime -4y=x

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Explain the existence and uniqueness of a solution of IVP consisting of xy′−4y=x6ex for the given initial condition y(0)=0.

Explain the existence and uniqueness of a solution of IVP consisting of xy′−4y=x6ex for the given initial condition y(0)=y0,y0>0.

Explain the existence and uniqueness of a solution of IVP consisting of xy′−4y=x6ex for the given initial condition y(x0)=y0,y0>0.

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Chapter 2 Solutions