If dx/dt = (9-x^2), solve the differential equation for x(t)

dx/dt = (9 - x²)
dx/dt = (3 - x)(3 + x)
dx / [ (3 - x)(3 + x) ] = dt

partial fractions
1 / [ (3 - x)(3 + x) ] = A/(3 - x) + B/(3 + x)
1 = A(3 + x) + B(3 - x)
1 = 3A + Ax + 3B - Bx

Ax = Bx
A = B

1 = 3A + 3B
1 = 3A + 3A
A = 1/6
B = 1/6


⅙ ∫ 1/(3 - x) dx + ⅙ ∫(3 + x) dx = ∫ dt
-⅙ ln(3 - x) + ⅙ ln(3 + x) = t + C
ln (3 + x)/(3 - x) = 6t + 6c
(3 + x)/(3 - x) = Ce^(6t)
(3 + x)/(3 - x) = Ce^(6t)
(3 + x) = (3 - x) Ce^(6t)
3 + x = 3Ce^(6t) - xCe^(6t)
x + xCe^(6t) = 3Ce^(6t) - 3
x [ 1 + Ce^(6t) ] = 3 [ Ce^(6t) - 1 ]

x = 3 [ Ce^(6t) - 1 ] / [ 1 + Ce^(6t) ]
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