Do you mean eliminate c_{1}, c_{2}, and c_{3} from this equation: y = c_{1}e^{5x} + c_{2}x + c_{3}? If so, this is my solution to it:
$y = c_1 e^{5x} + c_2 x + c_3$

$y' = 5c_1 e^{5x} + c_2$

$y'' = 25c_1 e^{5x}$

$y''' = 125c_1 e^{5x}$

$y''' - 5y'' = 0$ ← this is my answer.

Do you have answer key to this problem? If you do, update us.

Do you mean eliminate

c_{1},c_{2}, andc_{3}from this equation:y=c_{1}e^{5x}+c_{2}x+c_{3}? If so, this is my solution to it:$y = c_1 e^{5x} + c_2 x + c_3$

$y' = 5c_1 e^{5x} + c_2$

$y'' = 25c_1 e^{5x}$

$y''' = 125c_1 e^{5x}$

$y''' - 5y'' = 0$ ← this is my answer.

Do you have answer key to this problem? If you do, update us.