The solution of the differential equation sinx+cosxdy+cosx−sinxdx=0  is

(sin⁡x+cos⁡x)dy+(cos⁡x−sin⁡x)dx=0∫dy=∫sin⁡x−cos⁡xsin⁡x+cos⁡xdx Put cos⁡x+sin⁡x=t⇒y=−∫dtt=−ln⁡t+ln⁡c⇒y=ln⁡csin⁡x+cos⁡x⇒ey=csin⁡x+cos⁡x⇒ey(sin⁡x+cos⁡x)=c