This is first order non-linear ODE:
Solve the separable equation ( df(t))/( dt) = f(t)^2 + 1:
Divide both sides by f(t)^2 + 1:
(( df(t))/( dt))/(f(t)^2 + 1) = 1
Integrate both sides with respect to t:
integral(( df(t))/( dt))/(f(t)^2 + 1) dt = integral1 dt
Evaluate the integrals:
tan^(-1)(f(t)) = t + c_1, where c_1 is an arbitrary constant.
Solve for f(t):
Answer: |f(t) = tan(t + c_1)
