5.5-48

There are two terms in the hand solution, one of which is a special
case of y_h(x). Removing it leaves a single shortest term for y_p(x):
-(x/2)exp(-2x).

The answer can be tested from the maple function odetest() as follows:

        ODE := diff(y(x),x,x)-2*diff(y(x),x)-8*y(x)=3*exp(-2*x);
        sol := y(x)=(-1/2)*x*exp(-2*x);
        odetest(sol,ODE,y(x));


5.5-52

The answer obtained by integration by hand contains three terms,
two of which collect to (1/18)sin(3x)(sin^2(3x)+cos^2(3x)) which reduces
to (1/18)sin(3x), a special case of y_h(x). Since y_h(x) should not appear
in the shortest possible y_p(x), we remove it to obtain a single term
for the answer: -(x/6)cos(3x).



5.5-60

The problem statement has a typo but the book's answer in BOB is OK.
The formula for y_c given in the problem statement should be

   y_c = c1 x^(1/2) + c2 x^(3/2)

Use y1:=sqrt(x); and y2:=x*sqrt(x); when trying to do integrals of the
variation of parameters formula

                yp := y1*int(y2*(-f(x))/w,x) + y2*int(y1*f(x)/w,x);

where 

                 w:=y1*diff(y2,x)-y2*diff(y1,x);

Hint from Jon Engle, 10:45. If you got a coefficient of 72/7, then likely
you used the formula for y_c given in the problem, which means you should
change y_c to the one above.