First slide

Methods of solving first order first degree differential equations

Question

The solution of the differential equation $\frac{y + \sin x . \cos^{2} (x y)}{\cos^{2} (x y)} d x + (\frac{x}{\cos^{2} (x y)} + \sin y) d y = 0$ is

Difficult

A

$\tan x y + c o s x + \cos y = c$
B

$\tan x y - c o s x - \cos y = c$
C

$\tan x y + c o s x - \cos y = c$
D

none

Solution

Given equation can be rearranged as
$\begin{array}{l} \frac{y d x + x d y}{\cos^{2} (x y)} + \sin x d x + \sin y d y = 0 \\ \Rightarrow \frac{d (x y)}{\cos^{2} (x y)} + \sin x d x + \sin y d y = 0 \\ \int_{}^{} s e c^{2} (x y) d (x y) + \int \sin x d x + \int \sin y d y = 0 \\ \Rightarrow \tan x y - \cos x - \cos y = c \end{array}$

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