Solve the given differential equation by undetermined coefficients.
y''' − 3y'' + 3y' − y = ex − x + 21

Y = +   + + t - 18

Step-by-step explanation:

y''' − 3y'' + 3y' − y = ex − x + 21

Homogeneous solution:

First  we propose a solution:

Yh =

Y'h =

Y''h =

Y'''h =

Now we solve the following equation:

Y'''h - 3*Y''h + 3*Y'h - Yh = 0

- 3* + 3* - = 0

To solve the equation we must propose a solution to the  polynomial :

r = 1

To find the other r we divide the polynomial by (r-1) as you can see  

attached:

solving the equation:

(r-1)() = 0

= 0

r = 1

So we have 3 solution = 1

replacing in the main solution

Yh =   +   +

The t and is used because we must have 3 solution  linearly independent

Particular solution:

We must propose a Yp solution:

Yp =

Y'p =

Y''p =

Y'''p =

Y'''p - 3*Y''p + 3*Y'p - Yp =

=

equalizing coefficients of the same function:

- 12c_{1} = 0

9c_{1} = 0

3c_{1} = 0

c_{1} = 0

3c_{2} - c_{3} = 21 => c_{5} =

-c_{2} = -1

c_{2} = 1

c_{3} = -18

Then we have:

Y = +   + + t - 18

By pythagorean property,