To determine To show: The differential equation M + N y ′ = 0 , has an integrating factor of the form μ ( x y ) and then find a general formula for this integrating factor. Explanation Theorem used: “Let the functions M , N , M y , and N x , be continuous in the rectangular region R : α < x < β , γ < y < δ . Then M ( x , y ) + N ( x , y ) y ′ = 0 is an exact differential equation in R if and only if M y ( x , y ) = N x ( x , y ) at each point of R ”. Proof: It is given that ( ∂ N ( x , y ) ∂ x − ∂ M ( x , y ) ∂ y ) ( x M ( x , y ) − y N ( x , y ) ) = R , where R depends on the quantity xy only. By the theorem stated above, the equation is exact if and only if ∂ M ( x , y ) ∂ y = ∂ N ( x , y ) ∂ x . It is given that the equation M ( x , y ) + N ( x , y ) y ′ = 0 does not satisfy the above condition. Therefore, multiply the M ( x , y ) + N ( x , y ) y ′ = 0 by the integrating factor μ ( x , y ) . μ ( x , y ) M ( x , y ) + μ ( x , y ) N ( x , y ) d y d x = 0 The above equation is exact if ( μ ( x , y ) M ( x , y ) ) y = ( μ ( x , y ) N ( x , y ) ) x . ∂ μ ( x , y ) M ( x , y ) ∂ y = ∂ μ ( x , y ) N ( x , y ) ∂ x M ( x , y ) ∂ μ ( x , y ) ∂ y + μ ( x , y ) ∂ M ( x , y ) ∂ y = N ( x , y ) ∂ μ ( x , y ) ∂ x + μ ( x , y ) ∂ N ( x , y ) ∂ x M ( x , y ) ∂ μ ( x , y ) ∂ y − N ( x , y ) ∂ μ ( x , y ) ∂ x = μ ( x , y ) ∂ N ( x , y ) ∂ x − μ ( x , y ) ∂ M ( x , y ) ∂ y ∂ μ ( x , y ) ∂ x M ( x , y ) − ∂ μ ( x , y ) ∂ x N ( x , y ) = μ ( x , y ) ( ∂ N ( x , y ) ∂ x − ∂ M ( x , y ) ∂ y ) ...... ( 1 ) That is, the differential equation M ( x , y ) + N ( x , y ) y ′ = 0 is exact if it satisfies the above condition. Consider the given relation ( ∂ N ( x , y ) ∂ x − ∂ M ( x , y ) ∂ y ) ( x M ( x , y ) − y N ( x , y ) ) = R . ∂ N ( x , y ) ∂ x − ∂ M ( x , y ) ∂ y = R ( x y ) ( x M ( x , y ) − y N ( x , y ) ) ...... ( 2 ) From equation (1) and (2), M ( x , y ) ∂ μ ( x , y ) ∂ y − N ( x , y ) ∂ μ ( x , y ) ∂ x = μ ( x , y ) ( R ( x y ) ( x M ( x , y ) − y N ( x , y ) ) ) = R ( x y ) μ ( x , y ) x M ( x , y ) − R ( x y ) μ ( x , y ) y N ( x , y ) Compare the coefficient of M ( x , y )

10th Edition

ISBN: 9780470458327

Author: William E. Boyce, Richard C. DiPrima

Publisher: Wiley, John & Sons, Incorporated

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Chapter 2.6, Problem 24P

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To show: The differential equation M+Ny′=0, has an integrating factor of the form μ(xy) and then find a general formula for this integrating factor.

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