Does the solution of the following differential equation exists in some interval containing 0 and is the solution unique
$$(e^x\sin y)(y')^3+(e^x\cos y)y'+e^y \tan x=0,y(0)=0$$
how to find the solution of the this ODE? i have no idea to how to solve? what kind of differential equation it is is there special method to find solution of the this type of ODE
You use the implicit function theorem in the variable $v$ on $$ F(x,y,v)=(e^x\sin y)v^3+(e^x\cos y)v+e^y \tan x $$ where $F(0,0,0)=0$, $F_v(0,0,0)=1$ for the local existence of an explicit form $y'=f(x,y)$ and then the ODE theorems for a local solution.