Explanation Formula used: Formula for first order linear differential equation is, M ( x , y ) d x + N ( x , y ) d y = 0 If ∂ M ∂ y = ∂ N ∂ x then the differential equation is exact. For exact function F ( x , y ) , ∂ F ∂ x = M and ∂ F ∂ y = N General solution of the given differential equation is F ( x , y ) = C . Here, C is an arbitrary condition and F ( x , y ) = ∫ M ( x , y ) d x + g ( y ) . Calculation: Consider the given equation. [ sin ( x y ) + x y cos ( x y ) ] d x + [ 1 + x 2 cos ( x y ) ] d y = 0 Compare the above equation with the exact equations formula to check whether the given differential equation is exact or not as follows, Here, M ( x , y ) = sin ( x y ) + x y cos ( x y ) and N ( x , y ) = 1 + x 2 cos ( x y ) Now, find ∂ M ∂ y as follows, ∂ M ∂ y = ∂ ∂ y [ sin ( x y ) + x y cos ( x y ) ] = cos ( x y ) ∂ ∂ y ( x y ) + x ∂ ∂ y ( y cos ( x y ) ) = cos ( x y ) [ x ∂ ∂ y ( y ) ] + x [ y ∂ ∂ y cos ( x y ) + cos ( x y ) ∂ ∂ y ( y ) ] = cos ( x y ) x + x ( cos ( x y ) − x y sin ( x y ) ) = 2 x cos ( x y ) − x 2 y sin ( x y ) Find ∂ N ∂ x as follows, ∂ N ∂ x = ∂ ∂ x ( 1 + x 2 cos ( x y ) ) = ∂ ∂ x ( 1 ) + ∂ ∂ x ( x 2 cos ( x y ) ) = 0 + x 2 ∂ ∂ x ( cos ( x y ) ) + cos ( x y ) ∂ ∂ x ( x 2 ) = − x 2 y sin ( x y ) + 2 x cos ( x y ) Since, ∂ M ∂ y = ∂ N ∂ x Hence, the given differential equation is exact

Pearson eText Fundamentals of Differential Equations with Boundary Value Problems -- Instant Access (Pearson+)

7th Edition

ISBN: 9780137394524

Author: R. Nagle, Edward Saff

Publisher: PEARSON+

See similar textbooks

Question

Chapter 2.RP, Problem 5RP

To determine

To solve:

The equation [sin(xy)+xycos(xy)]dx+[1+x2cos(xy)]dy=0.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 2.RP, Problem 4RP

Chapter 2.RP, Problem 6RP

Students have asked these similar questions

T 1 7. Fill in the blanks to write the calculus problem that would result in the following integral (do not evaluate the interval). Draw a graph representing the problem. So π/2 2 2πxcosx dx Find the volume of the solid obtained when the region under the curve on the interval is rotated about the axis.

38,189 5. Draw a detailed graph to and set up, but do not evaluate, an integral for the volume of the solid obtained by rotating the region bounded by the curve: y = cos²x_for_ |x| ≤ and the curve y y = about the line x = =플 2 80 F3 a FEB 9 2 7 0 MacBook Air 3 2 stv DG

Find f(x) and g(x) such that h(x) = (fog)(x) and g(x) = 3 - 5x. h(x) = (3 –5x)3 – 7(3 −5x)2 + 3(3 −5x) – 1 - - - f(x) = ☐

Chapter 2 Solutions

Pearson eText Fundamentals of Differential Equations with Boundary Value Problems -- Instant Access (Pearson+)

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

Knowledge Booster

The equation [ sin ( x y ) + x y cos ( x y ) ] d x + [ 1 + x 2 cos ( x y ) ] d y = 0 .

To solve: The equation [sin(xy)+xycos(xy)]dx+[1+x2cos(xy)]dy=0.

Want to see the full answer?

Chapter 2 Solutions

To solve:

The equation [sin(xy)+xycos(xy)]dx+[1+x2cos(xy)]dy=0.