Question

check that if alpha is a real solution to the quadratic eq. x^2+bx+c = 0 , then y=e^(alpha*t) is a solution to the ordinary diff. eq. y''+by'+cy = 0

Answer 1

yeah, that's called characteristic equation

Answer 2

that doesn't really help me at all

Answer 3

you want to prove it right?

Answer 4

yeah but we haven't learned anything about characteristic equations

Answer 5

well ,

if you have differential equation of form
a y''+ b y' +c y=0

you can write it as algebraic equation and solve

a x^2 + b x^2 +c =0

Answer 6

why do we do that?

well , we assume soloution would be in form

y= e^at
so


y'=a e^at

y''= a^2 e^(at)

r y'' +s y'+t y =0

r (a^2 e^(at))+ s(a e^(at))+t(e^at)=0

divide by e^(a t)

r a^2 + s a+ t=0

Answer 7

instead of dividing both sides by e^(at) could you factor it out then realize that you end up with something that looks like the quadratic equation given in the earlier equation and since that =0 then multiplying e^(at) by 0 = 0

Answer 8

yes, that's better

Answer 9

thanks!