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af Əx Recall that a differential equation of the form M(x, y) dx + N(x, y) dy = 0 is exact if there is a region R of the xy-plane such that there exists a function f(x, y) defined on R such that = M(x, y) and af ду = N(x, y). By Theorem 2.4.1, the existence of the function f is equivalent to the conditions that M and N are continuous on a rectangular region a< x < b, c

af Əx Recall that a differential equation of the form M(x, y) dx + N(x, y) dy = 0 is exact if there is a region R of the xy-plane such that there exists a function f(x, y) defined on R such that = M(x, y) and af ду = N(x, y). By Theorem 2.4.1, the existence of the function f is equivalent to the conditions that M and N are continuous on a rectangular region a< x < b, c

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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af Recall that a differential equation of the form M(x, y) dx + N(x, y) dy = 0 is exact if there is a region R of the xy-plane such that there exists a function f(x, y) defined on R such that = M(x, y) and ?x af ду = N(x, y). By Theorem 2.4.1, the existence of the function f is equivalent to the conditions that M and N are continuous on a rectangular region a < x < b, c < y < d and the following partial derivatives are equal. ƏM ƏN ду ?х We are given a differential equation and must rewrite it in the form M(x, y) dx + N(x, y) dy = 0. x dy + y - Find the functions M and N. M(x, y) = = N(x, y) = xex. dy dx x dy = - 8xex - 6x² = 8xex - y + 6x² (8xex - y + 6x²) dx - 6x² dx = 0
Transcribed Image Text:af Recall that a differential equation of the form M(x, y) dx + N(x, y) dy = 0 is exact if there is a region R of the xy-plane such that there exists a function f(x, y) defined on R such that = M(x, y) and ?x af ду = N(x, y). By Theorem 2.4.1, the existence of the function f is equivalent to the conditions that M and N are continuous on a rectangular region a < x < b, c < y < d and the following partial derivatives are equal. ƏM ƏN ду ?х We are given a differential equation and must rewrite it in the form M(x, y) dx + N(x, y) dy = 0. x dy + y - Find the functions M and N. M(x, y) = = N(x, y) = xex. dy dx x dy = - 8xex - 6x² = 8xex - y + 6x² (8xex - y + 6x²) dx - 6x² dx = 0
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