Explanation Approach: 1. Suppose the first partial derivatives of M ( x , y ) and N ( x , y ) are continuous in a rectangle R , then, M ( x , y ) d x + N ( x , y ) d y = 0 is an exact equation in R if and only if the condition ∂ M ∂ y ( x , y ) = ∂ N ∂ x ( x , y ) holds for all ( x , y ) in R , otherwise M d x + N d y = 0 is not exact. 2. A first order equation is separable if it can be written in the form, d y d x = g ( x ) p ( y ) 3. An equation is linear if it can be written in the form, d y d x + P ( x ) y = Q ( x ) 4. If the equation, M ( x , y ) d x + N ( x , y ) d y = 0 is not exact, but the equation, μ ( x , y ) M ( x , y ) d x + μ ( x , y ) N ( x , y ) d y = 0 that results from multiplying by μ ( x , y ) is exact is called an integrating factor of the equation. Calculation: Consider the given expression. ( 2 x + y x − 1 ) d x + ( x y − 1 ) d y = 0 …… ( 1 ) Simplify equation ( 1 ) in the separable form. ( 2 x + y x ) d x = − ( x y − 1 ) d y ( 2 x 2 + y x ) d x = ( 1 − x y ) d y ( 2 x 2 + y x ( 1 − x y ) ) d x = d y d y d x = ( 2 x 2 + y x ( 1 − x y ) ) …... ( 2 ) Equation ( 2 ) cannot be written in a separable form of d y d x = g ( x ) p ( y ) . Hence, the given equation is not separable. Simplify equation ( 2 ) in linear form of d y d x + P ( x ) y = Q ( x ) . d y d x = ( 2 x 2 + y x ( 1 − x y ) ) d y d x = ( 2 x 2 x ( 1 − x y ) ) + ( y x ( 1 − x y ) ) d y d x = ( 2 x ( 1 − x y ) ) + ( y x ( 1 − x y ) ) d y d x − ( 2 x ( 1 − x y ) ) − ( y x ( 1 − x y ) ) = 0 …… ( 3 ) Equation ( 3 ) cannot be written in a linear form of d y d x = g ( x ) p ( y )

Fundamentals of Differential Equations and Boundary Value Problems

7th Edition

ISBN: 9780321977106

Author: Nagle, R. Kent

Publisher: Pearson Education, Limited

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Chapter 2.5, Problem 1E

To determine

To identify:

The equation (2x+yx−1)dx+(xy−1)dy=0 as separable, linear, exact, or having an integrating factor that is a function of either x alone or y alone.

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Chapter 2.4, Problem 36E

Chapter 2.5, Problem 2E

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Chapter 2 Solutions

Fundamentals of Differential Equations and Boundary Value Problems

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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The equation ( 2 x + y x − 1 ) d x + ( x y − 1 ) d y = 0 as separable, linear, exact, or having an integrating factor that is a function of either x alone or y alone.

To identify: The equation (2x+yx−1)dx+(xy−1)dy=0 as separable, linear, exact, or having an integrating factor that is a function of either x alone or y alone.

Want to see the full answer?

Chapter 2 Solutions

To identify:

The equation (2x+yx−1)dx+(xy−1)dy=0 as separable, linear, exact, or having an integrating factor that is a function of either x alone or y alone.