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Consider the ODE Mdx+Ndy=0 where M = M(x, y) and N = = N(x, y). (a) Show that has an integrating factor that depends only on the sum x + y if and only if the expression ƏN/дx – ӘМ/ду M-N depends only on x+y. Hint: In lecture, we found an integrating factor that depends only on x by assuming that the integrating factor had the form μ = u(x). For this problem, you want to assume that the integrating factor has the form μ = μ(x+y). (b) Use your result above to find an integrating factor for the ODE (3+y+xy)dx +(3+x+xy)dy = 0.

Consider the ODE Mdx+Ndy=0 where M = M(x, y) and N = = N(x, y). (a) Show that has an integrating factor that depends only on the sum x + y if and only if the expression ƏN/дx – ӘМ/ду M-N depends only on x+y. Hint: In lecture, we found an integrating factor that depends only on x by assuming that the integrating factor had the form μ = u(x). For this problem, you want to assume that the integrating factor has the form μ = μ(x+y). (b) Use your result above to find an integrating factor for the ODE (3+y+xy)dx +(3+x+xy)dy = 0.

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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Consider the ODE Mdx+Ndy=0 where M = M(x, y) and N= = N(x, y). (a) Show that has an integrating factor that depends only on the sum x + y if and only if the expression ƏN/дx – ӘМ/ду M-N depends only on x+y. Hint: In lecture, we found an integrating factor that depends only on x by assuming that the integrating factor had the form μ = u(x). For this problem, you want to assume that the integrating factor has the form μ = μ(x+y). (b) Use your result above to find an integrating factor for the ODE (3+y+xy)dx +(3+x+xy)dy = 0.
Transcribed Image Text:Consider the ODE Mdx+Ndy=0 where M = M(x, y) and N= = N(x, y). (a) Show that has an integrating factor that depends only on the sum x + y if and only if the expression ƏN/дx – ӘМ/ду M-N depends only on x+y. Hint: In lecture, we found an integrating factor that depends only on x by assuming that the integrating factor had the form μ = u(x). For this problem, you want to assume that the integrating factor has the form μ = μ(x+y). (b) Use your result above to find an integrating factor for the ODE (3+y+xy)dx +(3+x+xy)dy = 0.
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