Question

use the annihilator method to determinethe form of a particular solution for the given equation.

$y\text{'}\text{'}+2y\text{'}+y=x^2-x+1$

Short answer

$y_p\left(x\right)=c_3+c_{4}x+c_5{x}^2$

Step-by-step solution

Solve the homogeneous of the given equation

The homogeneous of the given equation is

$(D^2+2D+1)\left[y\right]=0$

The solution of the homogeneous is

$y_h\left(x\right)=c_1{e}^{-x}+c_{2}x{e}^{-x}$ (1)

Now $x^2-x+1$is annihilated by$D^3$

Then, every solution to the given nonhomogeneous equation also satisfies

.$D^3(D^2+2D+1)\left[y\right]=0$

Then

$y\left(x\right)=c_1{e}^{-x}+c_{2}x{e}^{-x}+c_3+c_{4}x+c_5{x}^2$ (2)

is the general solution to this homogeneous equation

We know$u\left(x\right)=u_h+u_p$

Comparing (1) & (2)

$y_p\left(x\right)=c_3+c_{4}x+c_5{x}^2$