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.`