(a) Explanation Given: The differential equation is d y d x = y 2 − x 2 x y and initial condition is y ( 1 ) = − 2 . Calculation: Consider the differential equation. d y d x = y 2 − x 2 x y (1) Consider the following substitution. y = u x (2) Differentiate equation (2) with respect to x . d y d x = u + x d u d x Substitute u x for y and u + x d u d x for d y d x in equation (1). u + x d u d x = ( u x ) 2 − x 2 x ( u x ) u + x d u d x = u 2 x 2 − x 2 u x 2 u + x d u d x = x 2 ( u 2 − 1 ) u x 2 x d u d x = ( u 2 − 1 ) u − u Further solving, x d u d x = ( u 2 − 1 ) − u 2 u x d u d x = − 1 u u d u = − d x x (3) Integrate equation (3) (b) To determine The largest interval I on which the solution is defined.

9th Edition

ISBN: 9781337292405

Author: ZILL

Publisher: CENGAGE L

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Differential Equations

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Chapter 2, Problem 36RE

(a)

To determine

To solve: The initial-value problem for implicit solution.

(b)

To determine

The largest interval I on which the solution is defined.

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Chapter 2, Problem 35RE

Chapter 2, Problem 37RE

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The initial-value problem for implicit solution.

Solution Summary: The author explains the implicit solution of the initial-value problem. The initial condition is y(1)=-sqrt.