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Following the Symbolic evaluation documentation, I am trying to solve the equation: $y'(x) = a*y(x), \quad y(0)=1$ where I later fix a via FindRoot, requiring that $y(2) = 2$:

ftest[unknown_?NumberQ] := First[y[2] /. NDSolve[{y'[x] == unknown*y[x], y[0] == 1}, y, {x, 0, 4}]] 
FindRoot[ftest[k] - 2, {k, 0, 4}]

This works well, but for some reason, replacing the "k" by "x" inside the Findroot outputs errors? I tried opening a new notebook and/or using Clear, but that did not solve the problem. Why does Mathematica treat x differently?

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You're using x within your ftest function, which is getting replaced by FindRoot. To fix this, you need to scope your ftest function. The following works:

ftest[unknown_?NumberQ] := 
 Module[{x}, 
  First[y[2] /. 
    NDSolve[{y'[x] == unknown*y[x], y[0] == 1}, y, {x, 0, 4}]]]
FindRoot[ftest[x] - 2, {x, 0, 4}]

If you read the error message your code outputs, you see something like NDSolve::dsvar: 0. cannot be used as a variable. FindRoot appears to be replacing the x in NDSolve[..., {x, 0, 4}] with 0, resulting in NDSolve[..., {0, 0, 4}], which is where it's trying to use 0 as a variable. Thus, you need to scope your inner x with a Module so that it doesn't get replaced.

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    $\begingroup$ The OP might note also that ftest[x] translates to the ODE y'[x] == x*y[x], which is quite a bit different than when unknown is a constant. However, in this case, as Carl points out, FindRoot replaces x by a number globally. This is what is meant by this line in the docs: "FindRoot...effectively uses Block to localize variables." $\endgroup$ Commented Jan 7, 2019 at 13:57
  • $\begingroup$ I see. I thought that inside function definition, the variables are automatically made local. Thank you very much for the reply. $\endgroup$ Commented Jan 7, 2019 at 15:32
  • $\begingroup$ @MichaelE2 Thanks a lot for the clarification. I don't have an extremely strong knowledge of the scoping mechanisms myself! $\endgroup$ Commented Jan 7, 2019 at 15:58

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