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

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

Answers (1)

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
\(dt= 2cos2y\)
\(int (cos2y)/(1+sin2y)dy= int (1/2dt)/t\)
\(= 1/2 int 1/t dt\) \(= 1/2 ln t+K\)
\(1/2 ln(1+sin2y)+k\)
\(int (cos2y)/(1+sin2y)dy= int(4x+2)/(sqrt(x2+x+3))dx\)
\(4 sqrt(x2+x+3)= 1/2 ln(1+sin2y)+c\)
0

Relevant Questions

asked 2020-11-27
Solve differential equation \(\displaystyle{x}{\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}}-{2}{y}={x}^{{3}}{{\sin}^{{2}}{\left({x}\right)}}\)
asked 2020-12-24
Solve differential equation \(\displaystyle{2}{x}{y}-{9}{x}^{{2}}+{\left({2}{y}+{x}^{{2}}+{1}\right)}{\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}}={0},\ {y}{\left({0}\right)}=-{3}\)
asked 2020-12-29
Solve differential equation \(2xy-9x^2+(2y+x^2+1)dy/dx=0\), y(0)= -3
asked 2020-11-16
Solve differential equation \((6x+1)y^2 dy/dx+3x^2+2y^3=0\), y(0)=1
asked 2021-02-25
Solve differential equation \(\displaystyle{\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}}+{\frac{{{y}}}{{{4}{x}}}}={3}{x},\ {y}{\left({2}\right)}={3}\)
asked 2021-03-07
Solve differential equation \(1+y^2+xy^2)dx+(x^2y+y+2xy)dy=0\)
asked 2020-11-14
Solve differential equation \(sin(x) dy/dx+(cos(x))y=0\), \(y((7pi)/6)=-2\)
asked 2020-11-27
Solve differential equation \(dx/dy+x/y= 1/(sqrt(1+y^2))\)
asked 2020-10-25
Solve differential equation \(dy/dx = e^4x(y-3)\)
asked 2021-02-05
Solve differential equation \(((3y^2-t^2)/y^5)dy/dx+t/(2y^4)=0\), y(1)=1
...