A) Find the two values of the unknown constant S for which x(t) = e^ St is a sol

Question

A) Find the two values of the unknown constant S for which x(t) = e^St is a solution of x'' + 6x' + 5x = 0.

Larger value of S=

Smaller value of S=

B) Consider the initial value problem x'' + 6x' + 5x = 0 with the initial condition x(0)=4 and x'(0)=-8. Find the constants c1 and c2 for which x(t) = c1 e-t + c2 e-5t satisfies the initial value problem.

c1=                                                        c2=

a. -5 b. 5    c. 1      d. -1        e. None of these

a. -5    b. 5    c. 1    d. -1    e. None of these

Explanation / Answer

a)

x'' + 6x' + 5x = 0

Characteristic eqn

^2 + 6 + 5 = 0

(+5)(+1) = 0

= -5, -1

x(t) = Ae-t + Be-5t

Larger S = -1

Smaller S = -5

b)

x(t) = = c1 e-t + c2 e-5t

x(0)=4 and x'(0)=-8

x(0) = 4 =c1 + c2

x'(t) = -c1 e-t -5c2 e-5t

x'(0) = -8 = -c1 -5c2

c1 = 3

c2 = 1

c)

x(t) = 3e-t + e-5t

x'(t) = -3e-t -5e-5t

limtx--> x'(t)/x(t) = (3e-t + e-5t) /(-3e-t -5e-5t)

limtx--> x'(t)/x(t) = (3e4t + 1) /(-3e4t -5) = -1

d)

x(t) = = c1 e-t + c2 e-5t

x(0)=1 and x'(0)=-5

x(0) = 1 =c1 + c2

x'(t) = -c1 e-t -5c2 e-5t

x'(0) = -5 = -c1 -5c2

c1= 0

c2 = 1

x'(t) = -5e-5t

x(t) = e-5t

limtx--> x'(t)/x(t) = -5e-5t /e-5t

= -5

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