Question

(x^2 3xy^2)dx (3x^2y 2y^2)dy=0求该全微分方程通解

Answer 1

答案为x³/3 + 3x²y²/2 + 2y³/3 = C

P = x² + 3xy²、P'y = 6xy
Q = 3x²y + 2y²、Q'x = 6xy
所以是恰当方程。
令原函数为u(x，y)
∂u/∂x = x² + 3xy²
u = ∫ (x² + 3xy²) dx = x³/3 + 3x²y²/2 + φ(y)
∂u/∂y = 3x²y + φ'(y) = 3x²y + 2y²
φ'(y) = 2y²
φ(y) = ∫ 2y² dy = 2y³/3 + C1
于是u(x，y) = x³/3 + 3x²y²/2 + 2y³/3 + C1
方程通解为u(x，y) = C
即x³/3 + 3x²y²/2 + 2y³/3 = C

Answer 2

有病