Explanation Approach: A differential equation that can be written in the form d y d x + p ( x ) y = q ( x ) y n , where n is a real constant, is called a Bernoulli equation. Calculation: The given equation y ′ + 4 x y = 4 x 3 y 1 / 2 is non linear. So, to convert it into linear equation, Multiply y − 1 / 2 on both sides in the above equation. y − 1 / 2 d y d x + 4 x y ( y − 1 / 2 ) = 4 x 3 y 1 / 2 ( y − 1 / 2 ) y − 1 / 2 d y d x + 4 x y 1 / 2 = 4 x 3 ... ( 1 ) Let u = y 1 / 2 , Differentiate u = y 1 / 2 with respect to x , d u d x = 1 2 y − 1 / 2 d y d x 2 d u d x = y − 1 / 2 d y d x Substitute 2 d u d x for y − 1 / 2 d y d x and u for y 1 / 2 in the equation (1). 2 d u d x + 4 x u = 4 x 3 d u d x + 2 x u = 2 x 3 ... ( 2 ) The standard linear differential equation is, d u d x + P y = Q Compare the differential equation (2) with standard linear differential equation. P = 2 x , Q = 2 x 3 The integrating factor is, I F = e ∫ P d x Substitute 2 x for P in the above equation. I F = e ∫ 2 x d x I F = e 2 ( x 2 2 ) I F = e ( x 2 ) I F = e x 2 The solution of linear equation is, u ( I F ) = ∫ Q ( I F ) d x Substitute 2 x 3 for Q and e x 2 for I F in the above equation

Differential Equations and Linear Algebra (4th Edition)

4th Edition

ISBN: 9780321964670

Author: Stephen W. Goode, Scott A. Annin

Publisher: PEARSON

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Chapter 1.8, Problem 46P

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The given differential equation y′+4xy=4x3y1/2.

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Chapter 1.8, Problem 45P

Chapter 1.8, Problem 47P

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Differential Equations and Linear Algebra (4th Edition)

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