Question

Use the method of indefinite coefficients and find a particle solution for differential equation: y '' - 5y '+ 5y = 3e ^ x.

Answer 1

Find a particle solution for differential equation in a form $y_p=a\cdot e^x,$ where a is unknown coefficient.

If $y_p=a\cdot e^x,$ then

• $y_p'=(a\cdot e^x)'=a\cdot e^x,$
• $y_p''=(a\cdot e^x)'=a\cdot e^x.$

Substitute these data into the differential equation:

$a\cdot e^x-5\cdot a\cdot e^x+5\cdot a\cdot e^x=3e^x,\\ \\a\cdot e^x=3e^x,\\ \\a=3.$

Thereofre, the particle solution for differential equation is

$y_p=3e^x.$