Question

如图，如何求这个线性微分方程通解公式的积分？

不难，但迷糊了

Answer 1

利用凑微分法可以求出结果。

Answer 2

y = e^[(-1/2)∫d(1+x^2)/(1+x^2)] { ∫xe^[(1/2)∫d(1+x^2)/(1+x^2)]dx + C }
= e^[(-1/2)ln(1+x^2)] { ∫xe^[(1/2)ln(1+x^2)]dx + C }
= [1/√(1+x^2)] [ ∫x√(1+x^2)dx + C ]
= [1/√(1+x^2)] [ (1/2)∫√(1+x^2)d(1+x^2) + C ]
= [1/√(1+x^2)] [ (1/3)(1+x^2)^(3/2) + C ]
= (1/3)(1+x^2) + C/√(1+x^2)