Example 6: Find a particular solution of .

Solution: Since a linear combination of and is of the form we have

> diff(a*x*sin(x)+b*(sin(x)-x*cos(x)),x,x)+a*x*sin(x)+b*(sin(x)-x*cos(x))-x*sin(x);

> p:=unapply(%,x);

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

No solution. Next, we try a particular solution of the form

.

> diff(x*(a*x*sin(x)+b*(sin(x)-x*cos(x))),x,x)+x*(a*x*sin(x)+b*(sin(x)-x*cos(x)))-x*sin(x);

> p:=unapply(%,x);

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

Hence .