Integrate a rule

Question

I want to implement a simple function to solve an exact differential equation, it's:

exact[Mx_, Ny_] := c + Integrate[g'[y], y] /. (Solve[
    D[Integrate[Mx, x] + g[y], y] == Ny, g'[y]])

Solve gives a rule: {{g'[y]-> something something}} and I want to Integrate the something something, but it just integrates g'[y] and gives as a result g[y].

Is there a way to force Mathematica to first apply the rule and then integrate?

Answer 1

Below is a long-form of the suggestion I showed in comments, where I am afraid I was not clear. I meant to suggest that you should do two things: 1. do the replacement within Integrate; 2. apply First to the result of Solve so you get an expression, rather than a single-element list containing your expression, from the replacement:

Clear[exact]
exact[Mx_, Ny_] := 
   c + Integrate[g'[y] /. First@Solve[D[Integrate[Mx, x] + g[y], y] == Ny, g'[y]], y]

Let's try it out on something simple:

exact[2 x + 2 y, 2 x + 2 y]

(* Out: c + y^2 *)

This seems to be what you wanted. Reconstructing piece by piece:

• Integrate[Mx, x] becomes Integrate[2 x + 2 y, x], which returns 2 (x^2/2 + x y);
• Add g[y] to get 2 (x^2/2 + x y) + g[y];
• Take the derivative wrt $y$ and set it equal to Ny: 2 x + g'[y] == 2 x + 2 y
• Solving for g'[y] returns {{g'[y] -> 2 y}}; take the first part, to give {g'[y] -> 2 y}

• Summarizing, you then apply the replacement within Integrate:

  g'[y] /. First@Solve[D[Integrate[2 x + 2 y, x] + g[y], y] == 2 x + 2 y, g'[y]]
  (* Out: 2 y *)
  

• Finally, find the antiderivative wrt $y$, which is y^2, and add c. This confirms the result obtained with your exact function.