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I want to solve a version of the Langevin's equation using RandomVariante with a normal distribution. Therefore, I used the next code:

f = NDSolve[{
DAF12'[t] == 1+Sqrt[2]RandomVariate[NormalDistribution[10, 5]]-DAF12[t],
DAF12[0] == 0},
DAF12[t],
{t, 0, 10}]

Then, a ploted the solution:

enter image description here

I was waiting much more variation in the solution, but it looks very smooth. So, I was wondering if each of the points of the solution takes the same random number. How can I know this? and How should I write the code in such a way that each point of the solution has a different random number?

J. M.'s missing motivation
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LINA MARCELA RUIZ GALVIS
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    Yes, the random number is evaluated once and that number is used throughout. Look up ItoProcess for instance or stochastic differential equations in the documentation/help center in Mathematica. – Michael E2 Aug 08 '20 at 19:53
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    It's possible to write it as an Itō or Stratonovich process but Mathematica will not necessarily solve it. For example: proc = ItoProcess[\[DifferentialD]X[ t] == (Sqrt[2] \[DifferentialD]W[t] - X[t]) \[DifferentialD]t, X[t], {X, 0}, t, W \[Distributed] WienerProcess[10, 5]] followed by RandomFunction[proc,{0,1,0.01}] will not produce any paths. It's possible I got something wrong though - have a look at these related questions and search for Langevin: https://mathematica.stackexchange.com/q/23183/72682 https://mathematica.stackexchange.com/q/23183/72682 – flinty Aug 08 '20 at 21:00

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