Question: replacement for OP command

As discussed in , the indices change some time at run time so previously working command is not working for me.

I am trying to use 

eq3 := applyop( u->Parts( u, phi[i](y) ), { 1,10,13,20 }, eq2 );

which clearly use the 'op' to get operands from eq2, but since the indices change at every run or on some run, the code is useless. The problem is that I have to integrate by parts some integrals of specfic structure and before this I just put there index and the command works perfectly alright but now its applying by parts integration on wrong integrals which are supposed to remain untouched.

How can I achieve this? 

To make myself simple , how  can I apply by parts integration on integrals having integrand like phi[i](y) * diff( w(y), y) without using op( rememeber eq3 uses 'op').


Here is my sheet: 

Many thanks!!


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