To determine To find: The general solution for given differential equation. Explanation Given: The differential equation 2 x y + x 3 = x d y d x Concept used: Integrating factor The function I ( x ) = e ∫ p ( x ) d x is called an integrating factor for the differential equation d y d x + P ( x ) y = Q ( x ) . Integration by parts of two functions u and v : ∫ u v d x = u ∫ v d x − [ ∫ d u d x ∫ v d x ] . where c is called the constant of integration. Calculation: Consider the differential equation, 2 x y + x 3 = x d y d x Divide above differential by x from both of the sides, 2 y + x 2 = d y d x Subtract both the sides by 2 y , d y d x − 2 y = x 2 (1) Compare above differential with d y d x + P ( x ) y = Q ( x ) . So, get the values P ( x ) = − 2 and Q ( x ) = x 2 . Now, find the integrating factor as I ( x ) = e ∫ p ( x ) d x I ( x ) = e − 2 ∫ d x = e − 2 x Multiply each term of equation (1) by I ( x ) e − 2 x d y d x − 2 e − 2 x y = x 2 e − 2 x The sum of terms on the left can now be replaced with D x ( y ⋅ e − 2 x ) D x ( y ⋅ e − 2 x ) = x 2 e − 2 x Integrating on both sides gives y ⋅ e − 2 x = ∫ x 2

Calculus With Applications, Books a la Carte Plus MyLab Math Package (11th Edition)

11th Edition

ISBN: 9780133886849

Author: Margaret L. Lial, Raymond N. Greenwell, Nathan P. Ritchey

Publisher: PEARSON

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Differential Equations

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Chapter 10.2, Problem 12E

To determine

To find: The general solution for given differential equation.

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Chapter 10.2, Problem 11E

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Calculus With Applications, Books a la Carte Plus MyLab Math Package (11th Edition)

Ch. 10.1 - Find all solutions of the differential equation . Ch. 10.1 - Prob. 2YT Ch. 10.1 - Prob. 3YT Ch. 10.1 - Prob. 4YT Ch. 10.1 - Prob. 1WE Ch. 10.1 - Prob. 2WE Ch. 10.1 - Prob. 3WE Ch. 10.1 - Prob. 4WE Ch. 10.1 - Prob. 5WE Ch. 10.1 - Find the general solution for each differential...

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