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))