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. To check that a DE M(x, y)dx + N(x, y)dy = 0) is exact, you check whether әM ду ON ?х Why? Give as much information as possible.

. To check that a DE M(x, y)dx + N(x, y)dy = 0) is exact, you check whether әM ду ON ?х Why? Give as much information as possible.

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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- To check that a differential equation (DE) \( M(x, y) dx + N(x, y) dy = 0 \) is exact, you check whether \( \frac{\partial M}{\partial y} = \frac{\partial N}{\partial x} \). Why? Give as much information as possible. This statement is related to the concept of exact differential equations in mathematics. An exact differential equation is one where the differential terms can be derived from a single function \( F(x, y) \) such that \( dF = M(x, y) dx + N(x, y) dy \). The condition \( \frac{\partial M}{\partial y} = \frac{\partial N}{\partial x} \) ensures that the mixed partial derivatives of \( F \) are equal, confirming the equation is exact.
Transcribed Image Text:- To check that a differential equation (DE) \( M(x, y) dx + N(x, y) dy = 0 \) is exact, you check whether \( \frac{\partial M}{\partial y} = \frac{\partial N}{\partial x} \). Why? Give as much information as possible. This statement is related to the concept of exact differential equations in mathematics. An exact differential equation is one where the differential terms can be derived from a single function \( F(x, y) \) such that \( dF = M(x, y) dx + N(x, y) dy \). The condition \( \frac{\partial M}{\partial y} = \frac{\partial N}{\partial x} \) ensures that the mixed partial derivatives of \( F \) are equal, confirming the equation is exact.
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