What is the solution for e^ycos(x)+cos(x)+(e^ysin(x)+e^y)y^'=0 ?

The solution for e^ycos(x)+cos(x)+(e^ysin(x)+e^y)y^'=0 is y=ln((c_{1})/(sin(x)+1)-1)

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e^ycos(x)+cos(x)+(e^ysin(x)+e^y)y^'=0

e^{y} cos (x) + cos (x) + (e^{y} sin (x) + e^{y}) y^{^{'}} = 0

y = ln (\frac{c _{1}}{sin ( x ) + 1} - 1)

Solution steps

e^{y} cos (x) + cos (x) + (e^{y} sin (x) + e^{y}) y^{'} (x) = 0

Solve separable ODE:

y = ln (\frac{c _{1}}{sin ( x ) + 1} - 1)

y = ln (\frac{c _{1}}{sin ( x ) + 1} - 1)

\frac{d y}{d x} = x^{3} y^{2}

x \frac{d y}{d x} = - y + \frac{1}{y ^{2}}

(x^{2} + e^{x}) d x + y d y = 0

x^{^{'}} = \frac{x ^{2} - 1}{2}

\frac{d y}{d x} = 3 x 2 y

What is the solution for e^ycos(x)+cos(x)+(e^ysin(x)+e^y)y^'=0 ?
The solution for e^ycos(x)+cos(x)+(e^ysin(x)+e^y)y^'=0 is y=ln((c_{1})/(sin(x)+1)-1)