Question

sinxsinydx + cosxcosydy = 0

Answer 1

SOLUTION :

TO SOLVE

$\sf{ \sin x \sin y \: dx + \cos x \cos y \: dy\: = 0 }$

EVALUATION

$\sf{ \sin x \sin y \: dx + \cos x \cos y \: dy\: = 0 }$

$\implies \sf{ \sin x \sin y \: dx = - \cos x \cos y \: dy\: }$

Dividing both sides by cos x sin y we get

$\sf{ \tan x \: dx = - \cot y \: dy\: }$

On integration we get

$\displaystyle\sf{ \int \tan x \: dx = - \int\cot y \: dy\: }$

$\implies \sf{ \ln | \sec x \: | = - \ln | \sin y| + \ln c \: }$

Where ln c is integration constant

$\implies \sf{ \ln | \sec x \: | + \ln | \sin y| = \ln c \: }$

$\implies \sf{ \ln | \sec x \: . \sin y| = \ln c \: }$

$\implies \sf{ | \sec x \: . \sin y| = c \: }$

RESULT

Hence the required solution is

$\boxed{ \sf{ \: \: | \sec x \: . \sin y| = c \: } \: \: }$

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Answer 2

Answer:

sinxsinydx + cosxcosydy = 0