Question

7.求微分方程(x2-y)dx-(x-y)dy=0的通解.(数1

Answer 1

(x^2-y)dx - (x-y)dy = 0
∂(x^2-y)/∂y = -1 = ∂[-(x-y)]/∂x, 是全微分方程。通解是
u(x,y) = ∫<0, x>(x^2-y)dx + ∫<0, y>[-(0-y)]dy
= x^3/3 - xy + y^2/2 = C

Answer 2

解:微分方程为(x²-y)dx-(x-y)dy=0，化为(x²-y)dx+(y-x)dy=0，由于∂(x²-y)/∂y=∂(y-x)/∂x=-1，则有x³/3-yx+g(y)=y²/2-xy+f(x)，得:g(y)=y²/2，f(x)=x³/3，微分方程的通解为x³/3-xy+y²/2=c(c为任意常数)