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. ( y 2 + 2 x y ) d x − x 2 d y = 0 …… ( 1 ) Simplify equation ( 1 ) in the separable form. ( y 2 + 2 x y ) d x = x 2 d y d y d x = ( y 2 + 2 x y ) x 2 …... ( 2 ) Equation ( 2 ) is not 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 = ( y 2 + 2 x y ) x 2 d x d y − ( y 2 + 2 x y ) x 2 = 0 …… ( 3 ) Equation ( 3 ) is not in a linear form of d y d x + P ( x ) y = Q ( x ) Hence, the given expression is not linear. Compare equation ( 1 ) with equation M d x + N d y = 0 . Therefore, M = ( y 2 + 2 x y ) N = − x 2 Partially differentiate M with respect to y . ∂ M ∂ y = ∂ ∂ y ( y 2 + 2 x y ) = ∂ ∂ y ( y 2 ) + 2 x ∂ ∂ y ( y ) = 2 y + 2 x ( 1 ) = 2 ( y + x ) Partially differentiate N with respect to x

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

7th Edition

ISBN: 9780137394524

Author: R. Nagle, Edward Saff

Publisher: PEARSON+

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Chapter 2.5, Problem 4E

To determine

To identify:

The equation (y2+2xy)dx−x2dy=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 2.5, Problem 3E

Chapter 2.5, Problem 5E

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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

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The equation ( y 2 + 2 x y ) d x − x 2 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 (y2+2xy)dx−x2dy=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 (y2+2xy)dx−x2dy=0 as separable, linear, exact, or having an integrating factor that is a function of either x alone or y alone.