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202y02mt1

202y02mt1 - B U Department of Mathematics Math 202...

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B U Department of Mathematics Math 202 Differential Equations Summer 2002 First Midterm This archive is a property of Bo˘ gazi¸ ci University Mathematics Department. The purpose of this archive is to organise and centralise the distribution of the exam questions and their solutions. This archive is a non-profit service and it must remain so. Do not let anyone sell and do not buy this archive, or any portion of it. Reproduction or distribution of this archive, or any portion of it, without non-profit purpose may result in severe civil and criminal penalties. 1. Show that the differential equation 2 xy 3 dx + (3 x 2 y 2 + x 2 y 3 + 1) dy = 0 , is not exact. Find an integrating factor and a one-parameter family of solutions for this equation. Solution: M = 2 xy 3 , N = 3 x 2 y 2 + x 2 y 3 + 1 M y = 6 xy 2 , N x = 6 xy 2 + 2 xy 3 M y = N x ; DE is not exact. N x - M y = 2 xy 3 = M An integrating factor μ = μ ( y ) exists: μ μ = N x - M y M = 1 μ ( y ) = e y Let ψ = ψ ( x, y ) be such that ψ x = 2 xy 3 e y , ψ y = (3 x 2 y 2 + x 2 y 3 + 1) e y ψ x = 2 xy 3 e y

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202y02mt1 - B U Department of Mathematics Math 202...

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