If the given solutions y⃗ 1(t)=[2e3t+4e−t3e3t+10e−t], y⃗ 2(t)=[−4e3t+2e−t−6e3t+5e−t]. form a fundamental set (i. e., linearly independent set) of solutions for the initial value problem y⃗ ′=[915−4−7]y⃗ , y⃗ (0)=[−30−55], impose the given initial condition and find the unique solution to the initial value problem. if the given solutions do not form a fundamental set, enter none in all of the answer blanks.

27 + 13x + 15 = (x + 5)( =

step-by-step explanation:

27 + 13x + 15 = (x + 5)( = 27 + 13x + 15 = (x + 5)( = 27 + 13x + 15 = (x + 5)( =

x^2 + 81

we can factor this using difference of square binomials.

rewrite the equation:

x^2 - (-81)

factor:

(a^2 - b^2) = (√a^2 + √b^2)(√a^2 - √b^2)

(x^2 - (-81) = (x + √-81)(x - √-81)

we can remove the - from under the radical by multiplying it by i.

(x^2 - (-81) = (x + i√81)(x - i√81)

(x^2 - (-81) = (x + 9i)(x - 9i)

we can check our answer by using foil:

x^2 - 9ix + 9ix - 81i^2

x^2 - 81^2

i^2 = -1

x^2 + 81