求微分方程y'-y/x+1=(x+1)3/2的通解

Question

求微分方程y'- y/x+1=(x+1)3/2的通解

Answer 1

你的微分方程是不是：y'-y/(x+1)=(x+1)^(3/2)
一阶线性微分方程,套公式
y=e^(∫1/(x+1)dx)[∫ (x+1)^(3/2)*e^(∫-1/(x+1)dx) dx + C]
=e^(ln(x+1))[∫ (x+1)^(3/2)*e^(-ln(x+1)) dx + C]
=(x+1)[∫ (x+1)^(3/2)*(x+1)^(-1) dx + C]
=(x+1)[∫ (x+1)^(1/2) dx + C]
=(x+1)[(2/3)(x+1)^(3/2) + C]
=(2/3)(x+1)^(5/2)+C(x+1)