微积分(A)随缘一题[15]

(1)

\[y'=\frac{2\arcsin x}{\sqrt{1-x^2}},y'(0)=0 \]

\[y''=2\frac{\frac{\sqrt{1-x^2}}{\sqrt{1-x^2}}-\arcsin x\frac{-2x}{2\sqrt{1-x^2}}}{1-x^2}=2\frac{1+\frac{x\arcsin x}{\sqrt{1-x^2}}}{1-x^2}=2\frac{1+xy'}{1-x^2} \]

所以：

\[(1-x^2)y''-xy'=2 \]

(2)

\[(-2x)y''+(1-x^2)y'''-y'-xy''=0 \\ (1-x^2)y'''-3xy''-y'=0 \]

\[(-2x)y'''+(1-x^2)y''''-3y''-3xy'''-y''=0 \\ (1-x^2)y''''-5xy'''-4y''=0 \]

\[-2xy''''+(1-x^2)y'''''-5y'''-5xy''''-4y'''=0 \\ (1-x^2)y'''''-7xy''''-9y'''=0 \]

不难得到：

\[y^{(n)}=\frac{(2n-3)xy^{(n-1)}+(n-2)^2y^{(n-2)}}{1-x^2} \]

所以：

\[y^{(n)}(0)=(n-2)^2y^{(n-2)}(0)=0 \]