Solution of an ODE in implicit form

Question

For some non-linear ODEs there is only implicit form of solution using DSolve. For example

DSolve[(y[x] + x - 1)*y'[x] - y[x] + 2 x + 3 == 0, y[x], x]

gives implicit solution

Solve[2/3 (Sqrt[2]ArcTan[(-2 + (2 (2 + 3 x))/(-1 + x + y[x]))/(2 Sqrt[2])]-Log[((-1 + x +y[x])^2 (3 + ((2 + 3 x) (-2 + (2 + 3 x)/(-1 + x +y[x])))/(-1 + x + y[x])))/(2 + 3 x)^2]) == C[1] + 4/3 Log[2 + 3 x], y[x]]

I'm solving a non-linear second-order ODE that has 4 explicit solutions. The formulas for these solutions are gigantic so I would like to see it in the implicit form because I hope it might be simpler.

Is there a way how to show the implicit form of solution of ODE even though the equation has the explicit solution?

(Just to be specific - the equation I'm solving reads

DSolve[(1 + G (A + y[x])^3) y''[x] + 3*G (A + y[x])^2 (y'[x])^2 + R == 0, y[x], x]

where $A,G,R \in \mathbb{R}$.)

Thank you for any kind of help.

Answer 1

I did not see there is an easy way to do it within DSolve. But for ODEs which could be integrated directly, using Integrate would be a possible choice to get the implicit solution. For the problem mentioned, it could be integrated directly by

Integrate[ 
  Integrate[ (1 + G (A + y[x])^3) y''[x] + 3G (A + y[x])^2 (y'[x])^2 + R, x] 
  - C[1], x] - C[2] == 0

(R x^2)/2 - x C[1] - C[2] + y[x] + A^3 G y[x] + 3/2 A^2 G y[x]^2 
 + A G y[x]^3 + 1/4 G y[x]^4 == 0

Here C[1] and C[2] are integration constants.