The general for Ricatti's equation:
y′=P(x)+Q(x)y+R(x)y2
Here
P(x)=2cosx2cos2x−sin2x;Q(x)=0;R(x)=2cosx1
Since y1=sinx is the particular solution of the equation, we can build the general solution of the equation in form
y=y1+w(x)1
Where w(x) is the solution to the first-order linear equation
w′=−(Q(x)+2R(x)y1)w−R(x)
or
w′=−cosxsinxw−2cosx1
w′+wtanx=−2cosx1
This is a first order linear equation, which solution is
w=c1cosx−2sinx
Therefore, the general solution of the equation is
y=sinx+Ccosx−sinx2,C=2c1
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