Example 5: Find a particular solution of .

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

> diff(a*exp(2*x),x,x)-4*diff(a*exp(2*x),x)+4*a*exp(2*x) - exp(2*x);

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

> diff(x*a*exp(2*x),x,x)-4*diff(x*a*exp(2*x),x)+4*x*a*exp(2*x) iff(x*a*exp(2*x),x)+4*x*a*exp(2*x) - exp(2*x);

Again no solution. Next, we try a particular solution of the form .

> diff(x^2*a*exp(2*x),x,x)-4*diff(x^2*a*exp(2*x),x)+4*x^2*a*exp(2*x) - exp(2*x);

> p:=unapply(%,x);

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

Hence .