CALCULUS: Solve the initial-value problem.

Solve the initial-value problem. (Assume the independent variable is x.)
25y'' + 40y' + 16y = 0, y(0) = 7, y'(0) = 0

Solve characteristic equation
25r² + 40r + 16 = 0
(5r + 4)² = 0
r = −4/5 (double root)

y = A e^(−4x/5) + B x e^(−4x/5)

Use initial conditions to find A and B

y(0) = 7
A e^0 + 0 = 7
A = 7

y' = −4/5 A e^(−4x/5) + B e^(−4x/5) − 4/5 B x e^(−4x/5)
y'(0) = 0
−4/5 (7) e^0 + B e^0 − 0 = 0
B = 28/5

y = 7 e^(−4x/5) + 28/5 x e^(−4x/5)
y = 7/5 e^(−4x/5) (4x + 5)

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