Explanation Approach: a) An equation is an exact equation when M ( x , y ) d x + N ( x , y ) d y = 0 and ∂ M ∂ y ( x , y ) = ∂ N ∂ x ( x , y ) holds for all ( x , y ) in R and if the compatibility condition is applicable. b) An equation is separable when it can be transformed into d y d x = g ( x ) p ( y ) . c) An equation is linear when it can be transformed into a 1 ( x ) d y d x + a 0 ( x ) y = b ( x ) , where a 1 ( x ) , a 0 ( x ) and b ( x ) depends on the independent variable x only. Calculation: Check for the exact equation. Consider the given differential equation. − 2 y − y 2 d x + ( 3 + 2 x − x 2 ) d y = 0 Compare the equation with the standard equation for the exact equation. M d x + N d y = 0 From the above comparison, M = − 2 y − y 2 N = 3 + 2 x − x 2 Check for the compatibility condition for the equation to be the exact equation. Take the partial differentiation of M with respect to y . ∂ M ∂ y = ∂ ∂ y − 2 y − y 2 = 1 2 − 2 y − y 2 ( − 2 − 2 y ) = − ( 1 + y ) − 2 y − y 2 Take the partial differentiation of N with respect to x . ∂ N ∂ x = ∂ ∂ x ( 3 + 2 x − x 2 ) = 2 − 2 x From the above, it is clear that the value of ∂ N ∂ x and ∂ M ∂ x are not equal. ∂ M ∂ y = ∂ N ∂ x − ( 1 + y ) − 2 y − y 2 ≠ 2 − 2 x Therefore, the equation − 2 y − y 2 d x + ( 3 + 2 x − x 2 ) d y = 0 is not an exact equation. Check for the separable. Represent the equation − 2 y − y 2 d x + ( 3 + 2 x − x 2 ) d y = 0 in the standard form of the separable equation

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ISBN: 9780321977175

Author: Nagle

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Chapter 2.4, Problem 3E

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The equation −2y−y2dx+(3+2x−x2)dy=0 is separable, linear, exact, or none of these.

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Chapter 2.4, Problem 2E

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To classify: The equation − 2 y − y 2 d x + ( 3 + 2 x − x 2 ) d y = 0 is separable, linear, exact, or none of these.

To classify: The equation −2y−y2dx+(3+2x−x2)dy=0 is separable, linear, exact, or none of these.

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Chapter 2 Solutions

To classify:

The equation −2y−y2dx+(3+2x−x2)dy=0 is separable, linear, exact, or none of these.