- #1
tracedinair
- 50
- 0
Homework Statement
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
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)
So,
(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.