Example 1: Find a particular solution of .

Solution: Since is a polynomial of degree 2, then we are looking for a particular solution that is a linear combination of and . This linear combination is of the form .

> diff(a*x^2+b*x+c,x,x)+a*x^2+b*x+c-x^2;

> p:=unapply(%,x);

> solve({p(0)=0,p(1)=0,p(2)=0},{a,b,c});

Therefore a particular solution is .

If , then we are looking for a particular solution that is a linear combination of and . This linear combination is of the form