Question

please help if an equation of a slope field is y'= (sinx)(siny) . what is the equation of the solution that passes through (0,npi), where n is any integer?

Answer 1

dy/dx= (sinx)(sin y)
dy/(siny)= dx/(sinx) now integrate .
-log(cosecy +coty)=-log (cosec x +cot x) -log c* ( put C=logc for easy calculation)
cosec y +cot y =c cosec x +c cot x
now plug the values cosec x , cot x not defined at 0.
and cosec y and cot y not defined at npi.
so there is not any real curve whose dy/dx=(sinx)(siny) and passes through (0 npi)