How to get partial expression from symbolic expression containing specific variable MAPLE?

Question

I have a symbolic expressions as below

y1 = (1/a)-(b/a^2)+x*a*b-x/b
y2 = a*b+a*x+b*sqrt(x)

now I need to get the partial expressions which

Answer 1

restart;

y1 := (1/a)-(b/a^2)+x*a*b-x/b:
y2 := a*b+a*x+b*sqrt(x):

K := (ee,x) -> `if`(ee::`+`,select(depends,ee,x),ee):

K( y1, x );

                     x
             x a b - -
                     b

K( y2, x );

                     (1/2)
            a x + b x     

#
# Leave alone an expression which is not a sum of terms.
#
K( sin(x+4)*x^3, x );

                         3
             sin(x + 4) x 

#
# Don't select subterms in which `x` is a just dummy name.
#
K( x^3 + sin(x) + Int(sqrt(x), x=a..b), x );

               3         
              x  + sin(x)

[edited]

y1 := (1/a)-(b/a^2)+x*a*b-x/b;

                      1   b            x
                y1 := - - -- + x a b - -
                      a    2           b
                          a             

op(3,y1);

                         x a b

depends(op(3,y1), x);

                          true

The select command maps its first argument over all the operands of its second argument.

select( s->depends(s,x), y1 );

                               x
                       x a b - -
                               b

A more terse syntax, where select maps its first argument depends over the operands of its second argument, and passes its third argument as additional options (to the selector).

select( depends, y1, x );

                               x
                       x a b - -
                               b

Now create a procedure to do it. Use a conditional test, so that it returns the first argument itself whenever that is not a sum of terms.

K1 := proc(ee, x)
  if type(ee,`+`) then
    select( depends, ee, x );
  else
    # leave it alone
    ee;
  end if;
end proc:

K1( y1, x);

                               x
                       x a b - -
                               b

Using a more terse syntax for that type-check.

K2 := proc(ee, x)
  if ee::`+` then
    select( depends, ee, x );
  else
    # leave it alone
    ee;
  end if;
end proc:

K2( y1, x);

                               x
                       x a b - -
                               b

Using a more terse syntax for that if..then..end if. This is the so-called operator form of if. The word if is within name-quotes, to distinguish it from the language keyword within an if...then...end if .

K3 := proc(ee, x)
  `if`( ee::`+` , select( depends, ee, x ), x );
end proc:

K3( y1, x);

                               x
                       x a b - -
                               b

Since the body of the procedure K3 has only a single statement then we can make it more terse, using the so-called operator form.

K4 := (ee, x) -> `if`( ee::`+` , select( depends, ee, x ), x ):

K4( y1, x);

                               x
                       x a b - -
                               b