Give the general solution to Can someone answer this questio

Give the general solution to...

(Can someone answer this question and explain the steps it took? Thank you.)

Given the general solution to y\'\' - y\' - 20y = 0.

Solution

Solution:

Option(D) from given choices:

y(x)=C₁ e ^5x + C₂  e ^-4x

Explanation:

The characteristic equation of the above is:

² --20=

factorize to get roots

²-5+4-20=0

(-5)+4(-5)=0

(+4)(-5)=0

-5=0 ------------>=5

+4=0 ----->=-4   

so₁₌ 5 and ₂=-4

are roots ,

General solution is given by:

y(x)=C₁e^₁x+C₂e^₂x

substituting roots we get

y(x)=C₁ e ^5x + C₂  e ^-4x