Question

Find the solution of Differential equation.

$\\ \implies \bf y \sin(2x) dx - (1+ {y}^{2} + { \cos}^{2}x)dy = 0$​

Answer 1

ANSWER :

$\implies \sf \dfrac {-1}{2} \: y \: cos \: 2x \: - \: \dfrac {3y}{2} \: - \: \dfrac {y^{3}}{3} \: = \: 0$

GIVEN :

$\\ \implies \bf y \sin(2x) dx - (1+ {y}^{2} + { \cos}^{2}x)dy = 0$

SOLUTION :

→ $\sf pdx \: + \: Qdx \: = \: 0$

→ $\sf y \: sin \: 2x \: dx \: + \: (-1 \: y^{2} \: - \: cos^{2}x) dy \: = \: 0$

→ $\sf \dfrac {dp}{dy} \: = \: sin \: 2x$

$\sf \dfrac {dQ}{dx} \: = \: - 2 \: cos \: x \times (-2 \: sin \: x) \: = \: 2 \: sin \: x \: cos \: x$

→ $\sf \int pdx \: = \: \dfrac {1}{2} y \: cos \: 2x \: + \:+ \: g(y)$

→ $\sf \int Qdy \: = \: - y \: - \: \dfrac {y^{33}}{3} \: - \: y \: cos^{2}x \: + \: h(x)$

→ $\sf cos^{2}x \: = \: \dfrac {1}{2} \: (1 \: + \: cos \: 2x)$

→ $\sf g(y) \: - \: y \: - \: \dfrac {y^{3}}{3} \: - \: \dfrac {y}{2} \: and \: h(x) \: = \: 0$

→ $\sf \dfrac {-1}{2} \: y \: cos \: 2x \: - \: \dfrac {3y}{2} \: - \: \dfrac {y^{3}}{3}$

$\implies \sf \dfrac {-1}{2} \: y \: cos \: 2x \: - \: \dfrac {3y}{2} \: - \: \dfrac {y^{3}}{3} \: = \: 0$

Answer 2

$\displaystyle\huge\red{\underline{\underline{Solution}}}$

FORMULA TO BE IMPLEMENTED

1. If u & v are two differentiable function then

$d(uv) = v \: du + u \: dv$

2.

$\displaystyle \int\limits_{}^{} x^{n} \, dx = \frac{ {x}^{n + 1} }{n + 1} + constant$

EVALUATION

$y \sin 2x \: dx - ( 1 + {y}^{2} + { \cos}^{2}x )dy = 0$

Multiplying both sides by (-1) we get

$- y \sin 2x \: dx + ( 1 + {y}^{2} + { \cos}^{2}x )dy = 0$

Rearranging we get

$( 1 + {y}^{2} )dy + ( { \cos}^{2}x \: dy \: - y \sin 2x \: dx ) = 0$

$\implies \: ( 1 + {y}^{2} )dy + \bigg[ { \cos}^{2}x \: dy \: + y \times ( - 2\sin x \cos x\: )dx \bigg ] = 0$

On integration

$\displaystyle \int\limits_{}^{} ( 1 + {y}^{2} )dy +\displaystyle \int\limits_{}^{} d(y \: { \cos}^{2}x ) = 0$

$\implies \: \displaystyle \: y + \frac{ {y}^{3} }{3} + y \: { \cos}^{2} x \: = c$

Where c is a constant

RESULT

SO THE REQUIRED SOLUTION IS

$\boxed{ \: \displaystyle \: y + \frac{ {y}^{3} }{3} + y \: { \cos}^{2} x \: = c}$