# Solve differential equation (cos^2y)/(4x+2)dy= ((cosy+siny)^2)/(sqrt(x^2+x+3))dx

Question
Solve differential equation $$(cos^2y)/(4x+2)dy= ((cosy+siny)^2)/(sqrt(x^2+x+3))dx$$

2021-03-08
$$(cos^2y)/(4x+2)dy= (cos^2y+sin^2y+2sinycosy)/(sqrt(x^2+x+3))dx$$
$$(cos^2y)/(4x+2)dy= (1+sin2y)/(sqrt(x^2+x+3))dx$$
$$(cos2y)/(1+sin2y)dy= (4x+2)/(sqrt(x^2+x+3))dx$$
Integrating on both sides
$$int (cos2y)/(1+sin2y)dy= int (4x+2)/(sqrt(x2+x+3))dx$$
$$int (4x+2)/(sqrt(x2+x+3))dx$$
$$Let x^2+x+3=u^2$$
$$2x+1dx= 2u du$$
$$4x+2dx= 4u du$$
$$int (4x+2)/(sqrt(x2+x+3))dx= (4u du)/u du$$
$$= 4du= 4u+c$$
$$= 4 sqrt(x^2+x+3)+c$$
$$int (cos2y)/(1+sin2y)dy$$
$$Let t=1+sin2y$$
Differentiating
$$dt= 2cos2y$$
$$int (cos2y)/(1+sin2y)dy= int (1/2dt)/t$$
$$= 1/2 int 1/t dt$$ $$= 1/2 ln t+K$$
$$1/2 ln(1+sin2y)+k$$
$$int (cos2y)/(1+sin2y)dy= int(4x+2)/(sqrt(x2+x+3))dx$$
$$4 sqrt(x2+x+3)= 1/2 ln(1+sin2y)+c$$

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