First order nonlinear differential equation

I've only learned how to solve first order linear differential equations, but this was on my homework.

y' = 10 - 2y^2

thanks!

y' = 10 - 2y^2
dy/dx = 2 * (5 - y^2)
dy / (5 - y^2) = 2 * dx

Let y = sqrt(5) * sin(t), then dy = sqrt(5) * cos(t) * dt

sqrt(5) * cos(t) * dt / (5 - 5 * sin(t)^2) = 2 * dx
sqrt(5) * cos(t) * dt / (5 * cos(t)^2) = 2 * dx
(1/sqrt(5)) * sec(t) * dt = 2 * dx
sec(t) * dt = 2 * sqrt(5) * dx

Integrate

ln(sec(t) + tan(t)) = 2 * sqrt(5) * x + C
sec(t) + tan(t) = C * e^(2 * sqrt(5) * x)

y = sqrt(5) * sin(t)
y^2 = 5 * sin(t)^2
y^2 = 5 * (1 - cos(t)^2)
y^2 = 5 - 5cos(t)^2
5cos(t)^2 = 5 - y^2
cos(t)^2 = (5 - y^2) / 5
sec(t)^2 = 5 / (5 - y^2)
sec(t) = sqrt(5 / (5 - y^2))

sec(t)^2 - tan(t)^2 = 1
5 / (5 - y^2) - 1 = tan(t)^2
tan(t)^2 = y^2 / (5 - y^2)
tan(t) = y / sqrt(5 - y^2)

sec(t) + tan(t) = (sqrt(5) + y) / sqrt(5 - y^2)

(sqrt(5) + y) / sqrt(5 - y^2) = C * e^(2 * sqrt(5) * x)

Let C * e^(2 * sqrt(5) * x) = m
Solve for y

(sqrt(5) + y) / sqrt(5 - y^2) = m
(5 + 2 * sqrt(5) * y + y^2) / (5 - y^2) = m^2
5 + 2 * sqrt(5) * y + y^2 = 5m^2 - m^2 * y^2
y^2 + m^2 * y^2 + 2 * sqrt(5) * y + 5 - 5m^2 = 0
y^2 * (1 + m^2) + y * (2 * sqrt(5)) + (5 - 5m^2) = 0
y = (-2 * sqrt(5) +/- sqrt(20 - 4 * (1 + m^2) * 5 * (1 - m^2))) / (2 * (1 +m^2))
y = (-2 * sqrt(5) +/- sqrt(20 - 20 * (1 - m^4)) / (2 * (1 + m^2))
y = (-2 * sqrt(5) +/- sqrt(20) * sqrt(1 - 1 + m^4)) / (2 * (1 + m^2))
y = (-2 * sqrt(5) +/- 2 * sqrt(5) * sqrt(m^4)) / (2 * (1 + m^2))
y = (-sqrt(5) +/- sqrt(5) * m^2) / (1 + m^2)
y = -sqrt(5) * (1 +/- m^2) / (1 + m^2)

2 solutions

y = -sqrt(5) * (1 + m^2) / (1 + m^2)
Which yields
y = -sqrt(5)
Not a good solution

y = -sqrt(5) * (1 - m^2) / (1 + m^2)

m = C * e^(2 * sqrt(5) * x)

y = -sqrt(5) * (1 - C * e^(4 * sqrt(5) * x)) / (1 + C * e^(4 * sqrt(5) * x))

2nd term calculus problem...separate the variables and integrate { partial fractions }