Unexpected warning from DSolve

Question

I'm trying to solve this differential equation with respect to $x$

$$ \frac{d^2y}{dx^2}-\bigg(\frac{dy}{dx}\bigg)^2-\frac{1}{x}\frac{dy}{dx}=\frac{1}{x^2} $$

With boundary condition $y(1)=2, y'(1)=-2$. Mathematica v10.0.0 was able to produce the correct solution, but with a warning that doesn't make much sense to me

Solve::incnst: "Inconsistent or redundant transcendental equation. After reduction, the bad equation is 1-C[1] == 0"

This is my code

 DSolve[{y''[x] - y'[x]^2 - y'[x]/x == 1/x^2, y[1] == 2, y'[1] == -2},y[x], x]
(*{{y[x] -> 2 - Log[x] - Log[1 + Log[x]]}}*)

Plus, if I substitute $1/x^2$ on the RHS with $a/x^2$, the solution by DSolve won't include the case where $a=1$.

 DSolve[{y''[x] - y'[x]^2 - y'[x]/x == a/x^2}, y[x], x]
 % /. a -> 1
(*{{y[x] -> C[2] + 1/2 (2 (-1 + Sqrt[1 - a]) Log[x] - 2 Log[x^(2 Sqrt[1 - a]) + C[1]])}}*)
(*{{y[x] -> C[2] + 1/2 (-2 Log[x] - 2 Log[1 + C[1]])}}*)

Is this a bug or should it be expected for DSolve?

Answer 1

I get the same message, and have gotten it before, on V10.2. I think it has to do with working out the boundary conditions when the solution & ODE has branch cuts. In this case there appears to be a singularity at a == 1, which needs to be approached via limits. Or so it seems to me.

One workaround is to use Reduce instead of Solve, as I did in Accessing Reduce from DSolve. We get no warning and a slightly different form of the solution.

With[{opts = Options[Solve]},
 Internal`WithLocalSettings[
  SetOptions[Solve, Method -> Reduce],
  {sola} = 
   DSolve[{y''[x] - y'[x]^2 - y'[x]/x == a/x^2, y[1] == 2, y'[1] == -2}, y, x],
  SetOptions[Solve, opts]
  ]
 ]
(*
  {{y -> Function[{x}, 
      2 + Log[1 + (-2 + 2 Sqrt[1 - a] + a)/a] - Log[x] + 
       Sqrt[1 - a] Log[x] - Log[(-2 + 2 Sqrt[1 - a] + a)/a + x^(2 Sqrt[1 - a])]]}}
*)

Check at a -> 1:

Limit[y[x] /. sola, a -> 1]
% // PowerExpand
(*
  2 + (I π)/2 - Log[x] - Log[I (1 + Log[x])]
  2 - Log[x] - Log[1 + Log[x]]
*)

This agrees with solution with a == 1:

With[{opts = Options[Solve]},
 Internal`WithLocalSettings[
  SetOptions[Solve, Method -> Reduce],
  DSolve[{y''[x] - y'[x]^2 - y'[x]/x == 1/x^2, y[1] == 2, y'[1] == -2}, y[x], x],
  SetOptions[Solve, opts]
  ]
 ]
(*
  {{y[x] -> 2 - Log[x] - Log[1 + Log[x]]}}
*)

Note: My experience shows that this approach does not always work when one gets a Solve::incnst message.

Answer 2

eqns1 = {y''[x] - y'[x]^2 - y'[x]/x == 1/x^2, y[1] == 2, y'[1] == -2};

soln1 = DSolve[eqns1, y, x][[1]] // Quiet

(*  {y -> Function[{x}, 2 - Log[x] - Log[1 + Log[x]]]}  *)

Verifing that soln1 satisfies eqns1

And @@ (eqns1 /. soln1 // Simplify)

(*  True  *)

Using your generalization of eqns1

eqns2 = {y''[x] - y'[x]^2 - y'[x]/x == a/x^2, y[1] == 2, y'[1] == -2};

soln2 = DSolve[eqns2, y, x][[1]] // Quiet

(*  {y -> Function[{x}, 
       2 + Log[2 - 2/a + 
             (2*Sqrt[1 - a])/a] - 
         Log[x] + Sqrt[1 - a]*Log[x] - 
         Log[(-2 + 2*Sqrt[1 - a] + a)/
               a + x^(2*Sqrt[1 - a])]]}  *)

Verifing that soln2 satisfies eqns2

And @@ (eqns2 /. soln2 // Simplify)

(*  True  *)

In the limit as a -> 1

soln21 = ReplacePart[soln2, {1, -1, -1} -> Limit[soln2[[1, -1, -1]], a -> 1]]

(*  {y -> Function[{x}, 2 + (I*Pi)/2 - 
         Log[x] - Log[I*(1 + Log[x])]]}  *)

Verifing that soln21 satisfies the original eqns1

And @@ (eqns1 /. soln21 // Simplify)

(*  True  *)

In the domain of x for which soln21 is real, soln21 equals soln1

(y[x] /. soln1) == ((y[x] /. soln21) // PowerExpand)

(*  True  *)

For the results to be real

Reduce[{x > 0, 1 + Log[x] > 0}, x]

(*  x > 1/E  *)

Plot[Evaluate[y[x] /. {soln1, soln21}],
 {x, 0, 2},
 PlotStyle -> {
   Directive[Thick, AbsoluteDashing[{7, 5}]],
   AbsoluteDashing[{10, 10}]},
 Epilog -> {Gray, Dashed,
   Tooltip[Line[{{1/E, 0}, {1/E, 6}}], x == 1/E]},
 PlotLegends -> Placed["Expressions", Bottom]]