Solve for the differential equation ycosxysinyxSolutionabove

Solve for the differential equation

y\'=(-cos(x)-y)/(sin(y)+x)

Solution

above d.e can be written as (dy/dx)(sin(y)+x) + (cos(x)+y)=0

Test for exactness d(cos(x)+y)/dy = 1= d(sin(y)+x)/dx

so the equation is exact.

dF/dx=cos(x)+y

dF/dy=sin(y)+x

integrating first d.e we get

F(x,y)=sin(x)+xy+A(y)

integrating second d.e we get

F(x,y)=-cos(y)+xy+B(x)

Equating both expressions for F(x,y) we get A(y)=-cosy and B(x)=sinx

F(x,y)=sinx-cosy+xy

solution is F(x,y)=K

sinx-cosy+xy=K