y′+xy/(1-x^2)-xy^1/2=0满足初始条件y(0)=0的特解

Question

y′+xy/(1-x^2)-xy^1/2=0 满足初始条件y(0)=0的特解

Answer 1

y′+xy/(1-x^2)-xy^1/2=0
除以y^1/2得：
y′/y^1/2+xy^1/2/(1-x^2)-x=0
设z=y^1/2,z'=(1/2)y′/y^1/2,代入得：
z'+2xz/(1-x^2)-2x=0,由一阶方程通解公式：
y^1/2=z= (1-x^2)(C+∫2x/(1-x^2)dx)
=(1-x^2)(C-ln|1-x^2|)
y(0)=0,C=0
y^1/2=(x^2-1)(ln|1-x^2|)