Question

(1 pt) If y=e5t is a solution to the differential equationd2ydt2−11dydt+ky=0,find the value of the constant k and the general solution to this equation.k= y= (Use constants A, B, etc., for any constants in your solution formula.)

Answer 1

Answer:

$y = A e^{5t} + B e^{6t}$

Step-by-step explanation:

given,

$y = e^{5t}$

and equation

y" - 11 y' + k y = 0...............(1)

now,

$y' = 5 e^{5t}$

$y"= 25 e^{5t}$

Putting value in equation (1)

$25 e^{5t} - 11(5 e^{5t}) + k e^{5t} = 0$

$- 30 + k = 0$

k = 30

now, differential equation becomes

y" - 11 y' +30 y = 0

( D² - 11 D + 30) y = 0

writing Auxiliary equation'

m² - 11 m + 30 = 0

(m - 5)(m-6) = 0

m = 5,6

now,

$y = A e^{5t} + B e^{6t}$