一阶线性微分方程
P=x² ,Q= 1/x
∫ P dx = 1/3 x³
y = e^(-x³/3) [ ∫ e^(x³/3) / x dx + C ]
∫ e^(x³/3) / x dx 不能用初等函数表示
所以原方程没有初等函数解
======以下答案可供参考======
供参考答案1:
y'+x^2y=x^(-1)
xy'+x^3y=1
x^3y'+x^5y=x^2
x^3dy+x^5ydx=x^2dx
x^3dy+(1/6)ydx^6=(1/3)dx^3
x^3dy+(1/3)x^3 ydx^3=(1/3)dx^3
x^3=uudy+(1/3)uydu=(1/3)du
dy+(1/3)ydu=(1/3)du/u
dy+(1/3)ydu=0
dy=(-1/3)ydu
y=C0e^(-u/3)
设y=C(u)e^(-u/3)
C'(u)e^(-u/3)=(1/3u)
C'(u)=(1/3u)e^(u/3)
C(u)=∫(1/3u)e^(u/3)du
通解y=[∫e^(u/3)du/(3u) ]*e^(-u/3)
