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Ordinary differential equations

1. The problem statement, all variables and given/known data

Show that φ(x) defined by,

(φ(x) - tan(x))/(φ(x) + cot(x)) = e^(∫(tan(x) + cot(x)) dx

is a solution of the differential equation y'(x) = 1 + y(x)^2

3. The attempt at a solution

Solving the right hand side first,

∫(tan(x) + cot(x) = ∫(tan(x)dx + ∫cot(x)dx = -ln|cos(x)| + ln|sin(x)|

e^(-ln|cos(x)| + ln|sin(x)|) = sin(x)/cos(x) = tan(x)


(φ(x) - tan(x))/(φ(x) + cot(x)) = tan(x)

And here's where I get stuck. I cannot solve for phi. I just end up getting lost in the algebra.
Assuming your work so far is correct, aside from the fact that the absolute values disappeared in the ln terms, and the constant of integration is missing, you have this:

(φ(x) - tan(x))/(φ(x) + cot(x)) = tan(x)
Multiply both sides by (φ(x) + cot(x)):
φ(x) - cot(x) = tan(x) * (φ(x) + cot(x))

Multiply the right side, and then get both terms involving φ(x) on one side and all other terms on the other side. Factor φ(x) from the terms containing it, and divide both sides by the other factor.

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