Two solutions for the same ODE

Question

I want to solve the following differential equation

$$ -\frac{1}{t}f'(t)-\frac{1}{2}f'(t)^2-f''(t)+\frac{1}{2t^2}=0 $$

for $t \in \mathbb{R}$. This can be done by hand. In particular, writing $g(t)= f'(t)$ and $g(t)=\nu(t)-1/t$, one can write the equation in the following form

$$ -\frac{1}{2}\nu(t)^2-\nu'(t)=0 $$

which has the solution $$\nu(t) = \frac{2}{t-k}$$ for some constant $k\in \mathbb{R}$. Therefore,

$$ f(t) = \int dt \left(\frac{2}{t-k}-\frac{1}{t}\right) = 2\ln|t-k|-\ln|t|+c $$ for some constant $c \in \mathbb{R}$.

Why is Mathematica returning instead $f(t) = c_2 + 2\ln(\cosh(\frac{1}{2}(2i c_1 + \ln(t)))$ with the following command?

-1/t f'[t] - 1/2 f'[t]^2 - f''[t] + 1/(2 t^2) == 0 // DSolve[#, f[t], t] &
(*Out:{{f[t] -> C[2] + 2 Log[Cosh[1/2 (2 I C[1] + Log[t])]]}} *)

EDIT

The solution returned by Mathematica seems to be different from $f(t)=2\ln|t-k|-\ln|t|+c$.

Answer 1

It is the same solution

ClearAll[f, t];
ode = -1/t f'[t] - 1/2 f'[t]^2 - f''[t] + 1/(2 t^2) == 0
sol = f[t] /. First@DSolve[ode, f[t], t]

sol = TrigToExp[sol]

sol = Simplify[sol]

But Exp[I C[1]] is constant, say C[3] then Exp[2 I C[1]] is C[3]^2. Therefore we can do this

sol /. {Exp[-I C[1]] -> 1/C[3], Exp[2 I C[1]] -> C[3]^2}

You used k for one constant and c for the second. The above used C[2] and C[3]

Maple gives

ode:= -1/t*diff(f(t),t) - 1/2*diff(f(t),t)^2 -diff(f(t),t$2) + 1/(2*t^2) = 0;
dsolve(ode)

$$ f \left( t \right) =\ln \left( {\frac { \left( -t{\it \_C2}+{\it \_C1 } \right) ^{2}}{4\,t}} \right) $$

Which if you want to work at it, will see it is the same.

To verify this in Mathemtica:

Simplify[ode /. 
  f -> Function[{t}, C[2] + 2*Log[(1 + t*C[3]^2)/(2*Sqrt[t]*C[3])]]]

(* True *)

Answer 2

foo = f[t]/. (-1/t f'[t] - 1/2 f'[t]^2 - f''[t] + 1/(2 t^2) == 0 // DSolve[#, f[t], t] &);

Simplify[TrigToExp[ foo //. {C[1] -> -(Log[-k]/(2 I)), C[2] -> c - 2 Log[Sqrt[-k/2]]}]]

(* c - Log[-2 k] + 2 Log[(-k + t)/(Sqrt[-k] Sqrt[t])] *)

In this latter form, the equivalence of the two solutions should be more obvious.