Question

Solution of the differential equation dy/dx=siny+x/sin2y-xcosy is

Answer 1

Answer:

the solution to the problem is solved below :-

Step-by-step explanation:

dy/dx=sin y+x/sin 2y-x cos y

dy/dx=sin y+x/2 sin y.cos y-x cos y

cos y dy/dx=sin y+x/2(sin y)-x

cos y dy/dx=(sin y/x)+1/2(sin  y/x)-1

put sin y/x=t

sin y=tx

cos y dy/dx=x dt/dx +t

x dt/dx+t=t+1/2t-1

x dt/dx =t+1/2t-1 -t=t+1-2t^2+t/2t-1=2*t-2t^2+1/2t-1

-2t^2+2t+1=v

-1/2 ln |  -2t^2-2t+1 | = ln |x|+c

-1/2 ln | -2(sin y/x)^2+2 sin y/2 +1 |=ln | x |+c

ln | x^2[-2sin ^2 y/x^2 + 2sin y/x +1] | = c1