求通解,dy/dx-3xy=xy^2

题目

求通解,dy/dx-3xy=xy^2

答案

dy/dx = xy² + 3xy
dy/dx = x(y² + 3y)
∫ 1/[y(y + 3)] dy = ∫ x dx
(1/3)∫ (3 + y - y)/[y(y + 3)] dy = ∫ x dx
∫ [1/y - 1/(y + 3)] dy = 3∫ x dx
lny - ln(y + 3) = 1.5x² + C
y/(y + 3) = Ce^(1.5x²)
或
Cye^(1.5x²) + 3Ce^(1.5x²) = y
y = 3/[Ce^(-1.5x²) - 1]