Answer to Question #172769 in Differential Equations for Yudhvir singh

Question

1 /xy + y is an integrating factor for the differential equation (x^2y + y^2)dx +( y^3-x^3)dy=0. True or false an

Answer 1

Solution

Let P(x,y) = x²y + y², Q(x,y) =  y³-x³, M(x,y) =  1 /xy + y.

If M(x,y) is an integrating factor then next condition is to be satisfied:

∂(M*P)/∂y = ∂(M*Q)/∂x

M*P = (x²y + y²)*(1 /xy + y) = x+y/x+x²y²+y³  =>  ∂(M*P)/∂y = 1/x+2x²y+3y² 

M*Q = (y³-x³)* (1 /xy + y) = y²/x-x²/y+y⁴-x³y  =>  ∂(M*Q)/∂x = -y²/x²-2x/y-3x²y

Therefore ∂(M*P)/∂y ≠ ∂(M*Q)/∂x.

So it’s False that M(x,y) =  1 /xy + y is an integrating factor for the differential equation (x²y + y²)dx +(y³-x³)dy=0.

Answer

False