Answer to Question #7428 in Differential Equations for noor
Is the following differential equation exact or not and solve it if its exact ?
( x^2 + cosy ) + ( y^2+x+sinx) dy/dx = x
Differential equation of the form I(x,y)dx + J(x,y)dy = 0 is exact only if dI/dy =dJ/dx . We can transform our equation to this form : ( x^2 + cosy )dx + ( y^2+x+sinx )dy = xdx ( x^2 - x + cosy )dx + ( y^2+x+sinx )dy = 0 . In this equation I(x,y) = x^2 - x + cosy and J(x,y) = y^2+x+sinx . Then we find : dI/dy = -siny and dJ/dx = 1 + cosx. We see that dI/dy not equal dJ/dx . That's why given equation is not exact.