First slide

Methods of solving first order first degree differential equations

Question

The solution of $(y (1 + x^{- 1}) + \sin y) d x + (x + \log_{e} x + x \cos y) d y = 0$ is

Moderate

A

$(1 + y^{- 1} . \sin y) + x^{- 1} . \log_{e} x = C$
B

$y + \sin y + x y \log_{e} x = C$
C

$x y + y \log_{e} x + x \sin y = C$
D

none of these

Solution

$\begin{array}{l} y d x + \frac{y}{x} \cdot d x + \sin y d x + x d y + \log_{e} x d y + x \cos y d y = 0 \\ \Rightarrow (y d x + x d y) + (y \cdot \frac{d x}{x} + \log_{e} x d y) + (\sin y d x + x \cos y d y) = 0 \\ \Rightarrow d (x y) + d (y \log_{e} x) + d (x \sin y) = 0 i n t e g r a t i n g o n b o t h s i d e s \\ x y + y \log_{e} x + x \sin y = C \end{array}$

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