Question: How did Maple obtain this solution that satisfies these initial conditions?

This is really a math question. But I can't figure how Maple did it.

Maple solves the ODE with 2 initial conditions correctly.

But if I use the general solution, and setup the set 2 equations, and tell Maple to solve _C1 and _C2, it says there is no solutions. Which is correct.

I would really like to know how Maple then managed to solve for these initial conditions.  This is problem from text book

I know how to solve it by hand to obtain general solution. But do not know how find solution that satisfies the IC's. when pluggin the initial conditions, the resulting 2 equation have no solution.


The goal is to now take the general solution found by Maple above, and manually solve for _C1 and _C2:

the_derivative:= diff(rhs(maple_general_sol),x):

Two equations, two unknown. But using solve or PDEtools:-Solve produce no solution.


Looking at equation (1) above, we see that it is the same as 

eq1:=1=(tan(_C2/(2*_C1))^2 + 1)/_C1^2;

Because 0=ln(Z) means Z=1. Using the above simpler equation now gives

eq1:=1=(tan(_C2/(2*_C1))^2 + 1)/_C1^2;
the_derivative:= diff(rhs(maple_general_sol),x):

We see that there is indeed no solution. Second equation above says _C1= tan(.). Let tan(.)=Z. Plugging this in first equation gives Z^2= (Z^1+1) which has no solution for Z since this says 0=1 

So direct appliction of the initial conditions produces no solution. So how did Maple find the solution it obtained? I also tried using limits. But no luck. Book does not show how to obtain solution either. 

Any ideas?


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