Explanation Given: [ 2 cos ( 2 x + y ) − x 2 ] d x + [ cos ( 2 x + y ) + e y ] d y = 0 , y ( 1 ) = 0 Approach: If ∂ M ∂ y = ∂ N ∂ x , then M ( x , y ) d x + N ( x , y ) d y = 0 is exact and there is a function F ( x , y ) such that ∂ F ∂ x ( x , y ) = M ( x , y ) ∂ F ∂ y ( x , y ) = N ( x , y ) The total differential of F ( x , y ) satisfies d F ( x , y ) = M ( x , y ) d x + N ( x , y ) d y which is equal to 0 . Calculation: [ 2 cos ( 2 x + y ) − x 2 ] d x + [ cos ( 2 x + y ) + e y ] d y = 0 is of the form M d x + N d y = 0 with M = [ 2 cos ( 2 x + y ) − x 2 ] and N = [ cos ( 2 x + y ) + e y ] ∂ M ∂ y = − 2 sin ( 2 x + y ) , ∂ N ∂ x = − 2 sin ( 2 x + y ) . As ∂ M ∂ y = ∂ N ∂ x , then the differential equation is exact. As d F = 0 ∂ F ∂ x d x + ∂ F ∂ y d y = 0 where ∂ F ∂ x = [ 2 cos ( 2 x + y ) − x 2 ] and ∂ F ∂ y = [ cos ( 2 x + y ) + e y ] Let ∂ F ∂ x = [ 2 cos ( 2 x + y ) − x 2 ] Integrate on both sides

Fundamentals Of Differential Equations And Boundary Value Problems Plus Mylab Math With Pearson Etext -- Title-specific Access Card Package (7th ... Fundamentals Of Differential Equations)

7th Edition

ISBN: 9780134768717

Author: R. Kent Nagle, Edward B. Saff, Arthur David Snider

Publisher: PEARSON

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Fundamentals Of Differential Equations And Boundary Value Problems Plus Mylab Math With Pearson Etext -- Title-specific Access Card Package (7th ... Fundamentals Of Differential Equations)

Ch. 2.2 - Prob. 1E Ch. 2.2 - In Problems 1-6, determine whether the given...Ch. 2.2 - Prob. 3E Ch. 2.2 - Prob. 4E Ch. 2.2 - Prob. 5E Ch. 2.2 - Prob. 6E Ch. 2.2 - Prob. 7E Ch. 2.2 - Prob. 8E Ch. 2.2 - Prob. 9E Ch. 2.2 - Prob. 10E

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