>> % Solution to Boyce-DiPrima problem 20, section 3.6 
>> dsolve('D2y + y = t*(1 + sin(t))')
 
ans =
 
sin(t)*C2+cos(t)*C1+t+1/4*cos(t)+1/4*t*sin(t)-1/4*cos(t)*t^2
 
 
>> % solution by undetermined coefficients
>> syms a b c d e f t
>> y = t + a*sin(t) + b*cos(t) + c*t*sin(t) + d*t*cos(t) ...
         + e*t^2*sin(t) + f*t^2*cos(t); 
>> diff(y, t, 2) + y
 
ans =
 
2*c*cos(t)-2*d*sin(t)+2*e*sin(t)+4*e*t*cos(t)+2*f*cos(t)-4*f*t*sin(t)+t
 
 
>> collect(ans, sin(t))
 
ans =
 
(-2*d-4*f*t+2*e)*sin(t)+2*c*cos(t)+t+4*e*t*cos(t)+2*f*cos(t)
 
 
>> collect(ans, cos(t))
 
ans =
 
(2*c+4*e*t+2*f)*cos(t)+(-2*d-4*f*t+2*e)*sin(t)+t
 
 
>> % Thus e = 0, c+f = 0, e-d = 0, -4*f = 1.
>> % a and b are arbitrary.
>> [csol, dsol, esol, fsol] = solve(e, c+f, e-d, 1+4*f, c, d, e, f);
>> [csol, dsol, esol, fsol]
 
ans =
 
[  1/4,    0,    0, -1/4]
 
 
>> subs(y, [c, d, e, f], [csol, dsol, esol, fsol])
 
ans =
 
t+a*sin(t)+b*cos(t)+1/4*t*sin(t)-1/4*cos(t)*t^2
 
 
>> % check:
>> diff(ans,t,2) + ans
 
ans =
 
t+t*sin(t)