y=y(x)由ln(x^2+y^2)=2arctan(y/x)决定,求d2y/dx2我做的和参考答案有细微差别,所以想请教一下正确的结果应该是多少?

Question

y=y(x)由ln(x^2+y^2)=2arctan(y/x)决定,求d2y/dx2
我做的和参考答案有细微差别,所以想请教一下正确的结果应该是多少?

Answer 1

ln(x^2+y^2)^1/2=arctan(y/x)
1/2ln(x^2+y^2)=arctan(y/x)
ln(x^2+y^2)=2arctan(y/x) 两边求导得
1/(x^2+y^2)*(2x+2yy')=2*1/(1+y^2/x^2)*(y'x-y)/x^2
(2x+2yy')/(x^2+y^2)=2x^2(y'x-y)/[(x^2+y^2)x^2]
2x+2yy'=2y'x-2y
2y'x-2yy'=2x+2y
y'=(x+y)/(x-y)