@Markiyan Hirnyk  yes, the next step. However, maybe someone who is expert in elliptic functions and Maple could derive it. 

@Carl Love 

Meanwhile I found a "solution".

Instead of  u₁'(z) or  D(u₁)(z)  the usage of

d
---    u₁(z)
dz

works well.               

@Alejandro Jakubi 

subs_problem_in_Mapl.mw

@Kitonum 

Your answer is ok, works.
In fact, my original problem is the following:

(Maple)

(Maple)

(Maple)

eq2

(Maple)

(Maple)

So u_2 is a function, but u_2 is unchanged in  D(u_2).
I need more help.
Thanks.

Thanks for your answers, work well.
To my problem evalc is better fitted.

@ecterrab 

Very nice!  Only two questions: Does the Maplesoft DifferentialEquation updates can handle the initial conditions given in Vector  form? 
Does it work with Maple 15?

Those Alejandro Jakubi mention are from 2010.

• What a surprise! One of my student asked the same. (He showed me corresponding Wolfram Math. commands and interested about the corresponding commands in Maple.)

Great! What I wanted. I prefer your first solution. Thanks.

Great! What I wanted. I prefer your first solution. Thanks.

Thanks. I see. Luckily the newer versions can handle this recursive problem.

Thanks. I see. Luckily the newer versions can handle this recursive problem.

@Markiyan Hirnyk Taking into account of the answer of acer it maybe a version problem.
Hopefully newer versions handle this problem properly.

Download MultiSeriesError.mw

I'm not an expert, but I think modifying temporarily Typesetting, interface, etc. somehow by Maple commands (demanding explicit use of multiplication symbol) could solve this problem.  I would like to
use it in a procedure.

I'm not an expert, but I think modifying temporarily Typesetting, interface, etc. somehow by Maple commands (demanding explicit use of multiplication symbol) could solve this problem.  I would like to
use it in a procedure.