Explanation Given: The differential equation is d y d x − 2 x y = − 1 with the initial condition y ( 0 ) = π 2 . Procedure used: The standard form of the first order linear differential equation with initial condition y ( x 0 ) = y 0 is, d y d x + P ( x ) y = Q ( x ) (1) To solve the equation (1) for y ( x ) , using the integrating factor method. The integrating factor of the differential equation is e ∫ P ( x ) d x . And the solution of the differential equation is, y ⋅ ( Integrating factor ) = ∫ ( ( Integrating factor ) ⋅ ( Q ( x ) ) ) d x + C (2) Here, C is constant. Calculation: The differential equation with initial condition y ( 0 ) = π 2 is, d y d x − 2 x y = − 1 (3) Compare equation (3) with equation (1). P ( x ) = − 2 x and Q ( x ) = − 1 Therefore the integrating factor of the differential equation (3) is, I . F = e ∫ − 2 x d x = e − x 2 Substitute e − x 2 for integrating factor and − 1 for Q ( x ) in equation (2). y ⋅ e − x 2 = ∫ ( e − x 2 × − 1 ) d x + C y ⋅ e − x 2 = − ∫ e − x 2 d x + C (4) Use the definite integration over the interval ( 0 , x ) . y ⋅ e − x 2 = − ∫ 0 x e − t 2 d t + C (5) Write the expression for the complementary error function. erf ( x ) = 2 π ∫ 0 x e − t 2 d t Substitute π 2 erf ( x ) for ∫ 0 x e − t 2 d t in equation (5)

A First Course in Differential Equations with Modeling Applications (MindTap Course List)

11th Edition

ISBN: 9781305965720

Author: Dennis G. Zill

Publisher: Cengage Learning

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Differential Equations

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Chapter 2.3, Problem 44E

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The solution of the given differential equation in the terms of erfc(x).

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Chapter 2.3, Problem 43E

Chapter 2.3, Problem 45E

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A First Course in Differential Equations with Modeling Applications (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 - In Problems 512 use computer software to obtain a...Ch. 2.1 - In Problems 512 use computer software to obtain a...Ch. 2.1 - In Problems 512 use computer software to obtain a...Ch. 2.1 - In Problems 512 use computer software to obtain a...Ch. 2.1 - In Problems 512 use computer software to obtain a...Ch. 2.1 - In Problems 512 use computer software to obtain a...

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The solution of the given differential equation in the terms of erfc ( x ) .

Solution Summary: The author explains the integrating factor method used to solve the differential equation for y(x).

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The solution of the given differential equation in the terms of erfc(x).

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