ANSWER
The particular solution is:

EXPLANATION
The given Ordinary Differential Equation is

The general solution to this Differential equation is:

To find the particular solution, we need to apply the initial conditions (ICs)

This implies that;




Hence the particular solution is

Any terminating decimal number is rational.
A rational number is a number that can be written as a fraction of integers.
0.444 = 444/1000 = 222/500 = 111/250
0.444 is the same as 111/250, a fraction of integers, so 0.444 is rational.
Let u = x.lnx, , w= x and t = lnx; w' =1 ; t' = 1/x
f(x) = e^(x.lnx) ; f(u) = e^(u); f'(u) = u'.e^(u)
let' find the derivative u' of u
u = w.t
u'= w't + t'w; u' = lnx + x/x = lnx+1
u' = x+1 and f'(u) = ln(x+1).e^(xlnx)
finally the derivative of f(x) =ln(x+1).e^(x.lnx) + 2x