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Math 2552 Practice Problems for final. Question 1. Please answer the following short questions and show all your work for full credit.
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Question 1. Please answer the following short questions and show all your work for full credit.
(1) Find the longest interval in which the solution of the initial value problem
(t − 0 .5)
dx dt
= tx + t^2 y
(cos t)
dy dt
= (sin t)x +
t − 2
y
x(1) = 1, y(1) = 2
is certain to exist.
(2) Is the function f (t) = et
3 of exponential order? Why?
(3) For the non-homogeneous equation y′′^ + 2y′^ + y = t^3 e−t, determine a suitable form for the particular solution if the method of underdetermined coefficients is to be used. Do not solve for the coefficients.
(4) Transform the following 4th order linear equation into a system of first order equations
y(4)^ − ty′′′^ + y′′^ = sin t.
Question 3. Find the general solution of
x
dy dx
− y = 2x^3 , x > 0.
Question 4. The tank initially contains 100 gal of fresh water. Water containing a salt concentration of sin t oz/gal flows into the tank at a rate of 2 gal/min, and the mixture in the tank flows out at the rate of 1 gal/min. Find the amount of salt in the tank at any time.
(2) Find the general solution of the following non-homogeneous system of differential equations
dx dt
− (^41) t (^241) t
x +
5 4 t
Question 6. Consider the system of equations
x′^ =
3 a
x.
(1) If a = 0, what is the type and stability of the critical point (0, 0)? Then find the solution to
the initial value problem with x(0) =
(2) In order for (0, 0) to be a spiral sink, what interval should a be in? Show your work. If it can never be a spiral sink, explain why.
Question 8. Compute the Laplace transform of the following functions:
(1) F (s) = (^) s (^2) +2^2 −ss+.
(2) F (s) = e
− 3 s s(s^2 +1).
Question 9. Use the Laplace transform to solve the following equation:
y(4)^ − y = 0
with the initial value y(0) = 1, y′(0) = 0, y′′(0) = 1, y′′′(0) = 0.
Question 11. Find the general solution of the following system of equations
x′^ =
(^) x.
Notice that λ^3 − 3 λ^2 + 4 = (λ + 1)(λ − 2)^2. You need to show your work about the computation of the determinant.
−cs
