(1+x^2)y''+2xy'=2x+1,求通解,麻烦答案写具体哈,

Question

(1+x^2)y''+2xy'=2x+1,求通解,麻烦答案写具体哈,

Answer 1

设t = dy/dx,dt/dx = d²y/dx²
(1 + x²) • dt/dx + 2tx = 2x + 1
(1 + x²) • dt/dx + t • d(x² + 1)/dx = 2x + 1
[t • (1 + x²)]' = 2x + 1
t • (1 + x²) = ∫ (2x + 1) dx = x² + x + C₁
t = (x² + x + C₁)/(1 + x²) = (x² + 1 + x - 1 + C₁)/(1 + x²) = 1 + (x - 1 + C₁)/(1 + x²)
dy/dx = ∫ dx + ∫ x/(1 + x²) dx + (C₁ - 1)∫ dx/(1 + x²)
y = x + (1/2)ln(x² + 1) + (C₁ - 1)arctan(x) + C₂