If y = e 4t is a solution to the differential equation d 2 y dt2 −10 dy dt +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

a. k = 24 b. y =

Step-by-step explanation:

a. Find the value of the constant k

Since   is a solution of d²y/dt² -10dy/dt + ky = 0, then   must satisfy the equation.

So, d²y/dt² =   and dy/dt =  

So, d²y/dt² -10dy/dt + ky = 0

- 10() + k() = 0

- + = 0

 - + = 0

- =  -

So, k = 24

b. Find the general solution to this equation.

Since k = 24, our equation is

d²y/dt² -10dy/dt + 24y = 0

The characteristic equation is thus

m² - 10m + 24 = 0

Factorizing this we have

m² - 6m - 4m + 24 = 0

m(m - 6) - 4(m - 6) = 0

(m - 6)(m - 4) = 0

m - 6 = 0 or m - 4 = 0

m = 6 or m = 4

Since we have real roots for the characteristic equation, then, the general solution is

y =

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step-by-step explanation:

step-by-step explanation:

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