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Which of the following statement(s) is (are) true: (a) The vector field F(x, y) = M(x, y)i + N(x, y)j is conservative if and only if the differential equation M(x, y)dx + N(x,y)dy = 0 is exact. (b) Let p(x, y) be an integrating factor of a non-exact differential equation M(x, y)dx + N(x, y)dy = 0. Then the vector filed G(x, y) = (x, y) M(x, y)i + (x, y)N(x, y)j is conservative. (c) If the vector filed G(x, y) = (x, y) M(x, y)i + (x,y)N(x, y)j is conservative, then the differential equation M(x,y) dx + N(x, y)dy = 0 is exact. (d) An integrating factor of the equation (sin x + cos x + e + y²)dx + y²dy = 0 is y².

Which of the following statement(s) is (are) true: (a) The vector field F(x, y) = M(x, y)i + N(x, y)j is conservative if and only if the differential equation M(x, y)dx + N(x,y)dy = 0 is exact. (b) Let p(x, y) be an integrating factor of a non-exact differential equation M(x, y)dx + N(x, y)dy = 0. Then the vector filed G(x, y) = (x, y) M(x, y)i + (x, y)N(x, y)j is conservative. (c) If the vector filed G(x, y) = (x, y) M(x, y)i + (x,y)N(x, y)j is conservative, then the differential equation M(x,y) dx + N(x, y)dy = 0 is exact. (d) An integrating factor of the equation (sin x + cos x + e + y²)dx + y²dy = 0 is y².

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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(10) Which of the following statement(s) is (are) true: (a) The vector field F(x, y) = M(x, y)i + N(x, y)j is conservative if and only if the differential equation M(x, y)dx + N(x, y)dy = 0 is exact. (b) Let p(x, y) be an integrating factor of a non-exact differential equation M(x, y)dx + N(x, y)dy = 0. Then the vector filed G(x, y) = (x, y) M(x, y)i + (x, y)N(x, y)j is conservative. (c) If the vector filed G(x, y) = (x, y) M(x, y)i + (x, y) N (x, y)j is conservative, then the differential equation M(x, y)dx + N(x, y)dy = 0 is exact. (d) An integrating factor of the equation (sin x + cos x + e + y²) dx + y²dy = 0 is y².
Transcribed Image Text:(10) Which of the following statement(s) is (are) true: (a) The vector field F(x, y) = M(x, y)i + N(x, y)j is conservative if and only if the differential equation M(x, y)dx + N(x, y)dy = 0 is exact. (b) Let p(x, y) be an integrating factor of a non-exact differential equation M(x, y)dx + N(x, y)dy = 0. Then the vector filed G(x, y) = (x, y) M(x, y)i + (x, y)N(x, y)j is conservative. (c) If the vector filed G(x, y) = (x, y) M(x, y)i + (x, y) N (x, y)j is conservative, then the differential equation M(x, y)dx + N(x, y)dy = 0 is exact. (d) An integrating factor of the equation (sin x + cos x + e + y²) dx + y²dy = 0 is y².
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