restart;

kernelopts(version);

A boundary value problem for an ODE

Let's solve the following boundary value problem:

Maple's dsolve() fails to find a solution:

Solution obtained by hand

The solution may be obtained by hand, as follows.

Integrate the DE once

.

where  is the integration constant.  Therefore

.

To determine the value of the constant , integrate the above over

and apply the boundary conditions:

whence
.

Having thus determined , we integrate the expression for  obtained above, and arrive at the solution
.

Remark: The solution obtained here is valid for any  as long as

the integrals encountered above make sense.  Note that there is
no requirement on the differentiability or even continuity of .

Technically,  can be any function such that  is integrable.

Wrong solution returned by dsolve()

Let us consider a special choice of the coefficient :

According the the calculations above, we have:

and therefore the correct solution is

Here is what the solution looks like:

Let's attempt to solve the same boundary value problem with Maple's dsolve().

That's very different from the correct solution obtained above.   To see the problem with it, let's recall that according to the DE, the expression should evaluate to   for some constant c,  but what we get through bad_sol is nothing like ;  in fact, it's not even continuous:

It appears that dsolve goes astray by attempting to expand the expression .   It shouldn't.

restart;

kernelopts(version);

A := piecewise(x<0, a[1], a[2]);

int(1/A, x=-1..1);

restart;

kernelopts(version);

plots:-display(
        plot(1, x=0..1),
         plot(1, x=2..3), thickness=5);

plots:-display(
        plot(1, x=0..1),
         plot(1, x=2..3),thickness=5, color=red);