The solution of the differential equation (x2sin3y−y2cosx)dx+(x3cosysin2y

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Question

The solution of the differential equation

(x^{2} s i n^{3} y - y^{2} c o s x) d x + (x^{3} c o s y s i n^{2} y - 2 y s i n x) d y = 0

is

Options:

A . x3sin3y=3y2sinx+C

B . x3sin3y+3y2sinx=C

C . x2sin3y+y3sinx=C

D . 2x2siny+y2sinx=C

Answer: Option A
:
A

(x^{2} s i n^{3} y - y^{2} c o s x) d x + (x^{3} c o s y s i n^{2} y - 2 y s i n x) d y = 0 \frac{d y}{d x} = \frac{y^{2} c o s x - x^{2} s i n^{3} y}{x^{3} c o s y s i n^{2} - 2 y s i n x} (x^{3} c o s y s i n^{2} y - 2 y s i n x) d y = (y^{2} c o s x - x^{2} s i n^{3} y) d x = 0 (\frac{x^{3}}{3} d s i n^{3} y - s i n d y^{2}) - s i n^{3} y d (\frac{x^{3}}{3}) + y^{2} d s i n x = 0

\frac{x^{3}}{3} d s i n^{2} y + s i n^{3} y d (\frac{x^{3}}{3}) - (s i n d y^{2} + y^{2} d s i n x)

d (\frac{x^{3}}{3} s i n^{3} y) - d (y^{2} s i n x) = 0 \frac{x^{3}}{3} s i n^{3} y - y^{2} s i n x = c

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