I am trying to model some physical phenomena using the following ODE : $f'' = 1/f$ with initial conditions $f(0) = 1/10 , f'(0) = 0$. I have solved it nummerically on a computer and also tried to linearize about a critical point. However there is no exact critical points and in the case the trajectory does not move the most is for points of the form $(M,0)$ where the larger the magnitude of M, the more sense it makes for the point to be almost critical. But these points are far away from my initial condition so I dont think it would even be worth linearizing about them. I also tried to guess a form for the solution, like a power function but I cant find any expression for f. Can anyone provide me a hand to find f?Ps: The actual equation I am trying to solve is the problem $f'' = 1/f -uf', u>0$ with the same initial conditions, but I thought maybe finding a closed form for the first problem could guide me through this more difficult one-
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