Checking an implicit function as a solution of a differential equation

Question

First of all hello and thanks in advance, and well my English is not that good so I'll try explain as good as I can the problem I'm having.

I want to verify if a function in implicit form is the solution of Differential equation. This is a problem of the Dennis G. Zill Book (A first Course in Differential Equations, Ninth Edition, page 10, problem 19). And the problem goes like this:

I have the following DE: $ x'(t)=(x-1)(1-2x) $ and i should check if the following implicit function:

$ \ln[(2x-1)/(x-1)]=t $

is a solution of it. I have done it by hand already, I just want to learn how to do this in Mathematica.

So far this is what i have:

prob19 = x'[t] == (x[t] - 1)*(1 - 2*x[t])

(*Derivative[1][x][t] == (1 - 2 x[t]) (-1 + x[t]) *)

Then

solprob19 = {{G -> Function[{x, t}, Log[E, (2*x - 1)/(x - 1)] == t]}}

(*{{G -> Function[{x, t}, Log[E, (2 x - 1)/(x - 1)] == t]}}*)

That is a guess I'm making on how to define the implicit function in a Pure Function way, based on:

• Checking the solution of a differential equation with a given solution

• Verification of the Solution

But when i do the ReplaceAll(/.) test I'm not getting {True} instead I'm getting this:

In[12]:= prob19 /. solprob19

Out[12]= {Derivative[1][x][t] == (1 - 2 x[t]) (-1 + x[t])}

With Simplify

In[13]:= Simplify[prob19 /. solprob19, {x > 1, t \[Epsilon] Real}]

Out[13]= {1 + 2 x[t]^2 + Derivative[1][x][t] == 3 x[t]}

So pretty much I'm just looking for pointers on how i could handle this problem, because i have failed in searching an answer for this.

Answer 1

One way is to solve for x and then follow the standard method

ode = x'[t] == (x[t] - 1) (1 - 2*x[t]);
myImplicitSolution = Log[(1 - 2 x)/(x - 1)] == t;
myExplicitSolution = x /. Solve[myImplicitSolution, x];
myExplicitSolution = x -> Function[{t}, Evaluate@myExplicitSolution];
Simplify[ode /. myExplicitSolution]

Answer 2

Here are a few ways that do not depend on being able to solve the implicit equation for the solution, based on differentiating the implicit equation. The key here is that the differentiated solution is linear in the derivative. Thus it can be used to eliminate the derivative from the ODE. If there is an initial condition, which is not given in the OP's example, it may be plugged into the solution and checked.

1. Using Eliminate reduces the ODE and differentiated solution to some measure-0 exceptions. These rule out conditions where the equations are undefined (due to division by zero or taking the logarithm of zero).

ode = x'[t] == (x[t] - 1) (1 - 2 x[t]);
implsol = Log[(2*x[t] - 1)/(x[t] - 1)] == t;
Eliminate[{ode, implsol}, {x'[t]}]
(*  -1 + x[t] != 0 && -1 + 2 x[t] != 0  *)

2. If we use Solve, the solution should be generically true, which is indicated by the output being {{}}.

Solve[{ode, implsol}, x[t], {x'[t]}]

Solve::fulldim: The solution set contains a full-dimensional component; use Reduce for complete solution information. >>

(*  {{}}  *)

3. Finally, we can rather straightforwardly solve the differentiated solution for the derivative and plug it into the ODE.

ode /. Solve[D[implsol, t], {x'[t]}] // Simplify
(*  {True}  *)

4. As an alternative to (3), one can check whether the residual will simplify to zero. When you get a warning about inverse functions, branch cuts, and so forth, one might wish to check numerically as I did in DSolve for Second Order Differential to eliminate one of the solutions returned by DSolve. That involves plugging into the residual random numbers for any parameters, the independent variable, and the dependent variable(s) and their derivatives. For a solution to the ODE, one should expect errors on the order of round-off error. In the present case, the residual simplifies symbolically to zero, which needs no numerical checking:

residual[a_ == b_] := a - b;
residual@ode /. Solve[D[implsol, t], {x'[t]}] // Simplify
(*  {0}  *)

If it simplifies to a function, one might try FullSimplify or substituting random values for the independent variable t. It is possible in some cases that due to branch cuts, the solution is invalid only on some interval, which would probably prevent the expression from simplifying down to 0.

---

More examples

This ODE is from the docs for DSolve, and DSolve returns an implicit solution in terms of Solve.

ode = y'[x] == y[x]^3 - ((x + 1) y[x]^2)/x;
dsol = DSolve[ode, y[x], x]
implsol = First@dsol          (* should check that dsol is of the form Solve[..] *)
(*
  Solve[E^(-x + 1/y[x])/x + C[1] + ExpIntegralEi[-x + 1/y[x]] == 0, y[x]]
  E^(-x + 1/y[x])/x + C[1] + ExpIntegralEi[-x + 1/y[x]] == 0
*)

Methods 1-3:

Eliminate[{ode, D[implsol, x]}, y'[x]]
Solve[{ode, D[implsol, x]}, y[x], {y'[x]}]
ode /. Solve[D[implsol, x], {y'[x]}] // Simplify
(*
  x != 0 && y[x] != 0 && -1 + x y[x] != 0  (* 1 *)
  {{}}                                     (* 2 *)
  {True}                                   (* 3 *)
*)

Here is another example from Solution of an ODE in implicit form:

ode = (y[x] + x - 1)*y'[x] - y[x] + 2 x + 3 == 0;
DSolve[ode, y[x], x];
implsol = First@%;

Eliminate[{ode, D[implsol, x]}, y'[x]]
Solve[{ode, D[implsol, x]}, y[x], {y'[x]}]
ode /. Solve[D[implsol, x], {y'[x]}] // Simplify
(*
  11 + 8 x + 6 x^2 - 10 y[x] + 3 y[x]^2 != 0
*)

Solve::fulldim: The solution set contains a full-dimensional component; use Reduce for complete solution information.

(*
  {{}}
  {True}
*)

Answer 3

This your differential equation:

prob19 = x'[t] == (x[t] - 1)*(1 - 2*x[t])

and this your implicit solution:

sol19 = Log[(2*x[t] - 1)/(x[t] - 1)] == t;

Then:

Simplify[prob19, D[sol19, t]]

(* True *)

Of course, this command internally uses the techniques given in Michael's answer.