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Here is a MWE for the problem I have:

NDSolve[{y'[x] == y[x] + x + z[x], y[2] == 1}, y, {x, 2, 4}]

where z is determined by

z x^z Log[z] == 1

which can't be solved for z[x] in simple form.

One possible way is to write ODE for z[x] also and solve the coupled differential equations. But for the actual problem at hand, that takes too long. Any suggestions for a different method to solve for y[x]?

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The DAE-functionality of NDSolve can solve your problem:

sol = NDSolve[{y'[x] == y[x] + x + z[x], y[2] == 1,z[x] x^z[x] Log[z[x]] == 1}, {y, z}, {x, 2, 4}]
Plot[{y[x], z[x]} /. sol, {x, 2, 4}]   

enter image description here

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    $\begingroup$ +1 Recommend that you use Evaluate, i.e., Plot[Evaluate[{y[x], z[x]} /. sol], {x, 2, 4}] $\endgroup$ – Bob Hanlon Apr 4 '19 at 18:09

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