I think I found it but I'd be more than happy to look at other alternatives if any provided: Shift + Ctrl + O to open Options > Notebook options > Evaluation options > Global preferences from the ...

Vitaly's answer is correct in that it fantastically produces a splined random disturbace surface. However, I was unable to use it as an initial condition for my NDSolve[...]. Based on whuber's ...

Replacing N[a] with SetPrecision[a,5] helped. The above answer by Fred Daniel Kline helps too but isn't really what I am looking for.

First solve this: a = NIntegrate[Sqrt[1 + E^tt^2 Sin[tt^2]], {tt, 0, 1}] And then do this: DSolve[{D[u[t], t] == a u[t], u == 1}, u[t], t] That should solve the problem.

This problem seems to be really stiff as whuber has pointed out. Using the BDF method and allowing NDSolve to ruminate about the order, didn't quite change the solution (z value where stiffness was ...

As prompted by one of the comments, stiffness switching did help some. I usually go with Method->LSODA for stiff equations but tried Method -> {"StiffnessSwitching", Method -> {"...

As late as this answer may be: sol = NDSolve[{f'''[η] + 0.5 f[η] f''[η] == 0, f == f' == 0, f' == 1}, f, η] Plot[f[η] /. First[sol], {η, 0, 10}] Plot[f'[η] /. First[sol], {η, 0, 10}] ...