Please show all the steps Given that yt c1e10t c2e10t is a

Please show all the steps

Given that y(t) = c_1e^10t + c_2e^-10t is a solution of the equation y\" - 100y = 0, where c_1 and c_2 are arbitrary constants, find a function y(t) that solves the initial value problem: y\" - 100y = 0, y(0) = 10, y\'(0) = 100.

Solution

Given:

y(t) = c₁ e^10t + c₂ e^-10t    (1)

(1) is a solution of the equation:

y\'\' - 100 y = 0                 (2).

Initial conditions:

y(0) = 10                      (3)

y\'(0) = 100                   (4)

Putting t = 0 in (1) & using (3), we get:

c₁ + c₂ = 10              (5).

Differentiating (1), we get:

y\'(t) = 10 c₁ e^10t - 10 c₂ e^-10t         (6)

Putting t = 0 in (6) & using (4), we get:

10c₁ - 10 c₂ = 100

So,

c₁ - c₂ = 10            (7)

(5)+(7) gives:

2c₁ = 20

So,

c₁ = 10

Sustituting in (7), we get:

c₂ = 0.

So, the required solution of the initial value problem is:

y(t) = 10 e^10t