(1 + y^2) / y' + x·y = 0
y' · y / (- y^2 - 1) = 1 / x
dy/dx · y / (- y^2 - 1) = 1/x
y / (- y^2 - 1) dy = 1/x dx
∫ -y / (y^2 + 1) dy = ∫ 1/x dx
Subst.
z = y^2 + 1
1 dz = 2·y dy
dy = dz / (2·y)
∫ -y / z dz / (2·y) = ∫ 1/x dx
∫ -1 / (2·z) dz = ∫ 1/x dx
- 1/2·LN(z) = LN(x) + C
Resubst.
- 1/2·LN(y^2 + 1) = LN(x) + C1
LN(y^2 + 1) = -2·LN(x) + C2
LN(y^2 + 1) = LN(x^{-2}) + C2
y^2 + 1 = e^{C2} / x^2
y^2 = C3 / x^2 - 1
y^2 = (C3 - x^2) / x^2
y = ± √(C3 - x^2) / x
Vereinfachung der Konstanten
y = ± √(C - x^2) / x