How can get two solution of the ODE with degree two ?

Questions
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1

i try to get the same result as the paper did with other platform too but i can’t get the same solution?
any one know how can get thus solution

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2

It does not seem to me that equation (17) is a general solution of (14):

eq = y'[x]^2 == a*y[x]^2 + b*y[x]^3 + c*y[x]^4;
sol = Function[x, (-a b Sech[Sqrt[a]/2 x]^2)/(
   b^2 - a c (1 + \[Epsilon] Tanh[Sqrt[a]/2 x])^2)];
eq /. y -> sol // Simplify
% /. x -> 0 // Simplify
3

so why he did that there is any way for get the same solution ?
because i need the same solution to comparing my method to his method

4

Equations (17) and (18) are solutions of equation (14) only when \[Epsilon] == \[PlusMinus]1, and they are not the general solutions, but only particular solutions. Formula (18) has a removable singularity at zero. DSolve gives you the general solution in terms of the parameter C[1]. You can recover solution (17) by finding which value of C[1] gives (17):

{genSol1, genSol2} = y /.
   DSolve[y'[x]^2 ==
     a*y[x]^2 + b*y[x]^3 + c*y[x]^4, y, x];
sol17 = Function[x, (-a b Sech[Sqrt[a]/2 x]^2)/
    (b^2 - a c (1 + \[Epsilon] Tanh[Sqrt[a]/2 x])^2)];
Simplify[genSol1[x] - sol17[x] /. \[Epsilon] -> 1 /. x -> 0];
parameterValues = Solve[% == 0, C[1]] /. C[2] -> 0
genSol1[x] == sol[x] /. \[Epsilon] -> 1 /. 
  parameterValues[[-1]] // Simplify