Find the solution for (D^2y)+(2Dy)-24y=0; Dy = 7, y=0, x=0

Question

(D^2y)+(2Dy)-24y=0; Dy = 7, y=0, x=0

Answer 1

Find the solution for (D^2y)+(2Dy)-24y=0; Dy(0) = 7, y(0)=0.

Auxiliary equation

m^2 + 2m - 24 = 0

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

m = -6, m = 4

Hence the general solution is: y(x) = A.e^(-6x) + B.e^4(x)

And y'(x) = -6A.e^(-6x) + 4B.e^(4x)

Initial conditions

y(0) =  0 = A + B

y'(0) = 7 = -6A + 4B

Substituting for A = -B from the 1st equation into the 2nd equation,

7 = 6B + 4B = 10B

Hence B = 7/10 and A = -7/10

Solution is: y(x) = (7/10)(e^(4x) - e^(-6x))