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Differential Equations Final Exam Practice
Solutions
- A tank originally contains 10 gal of water with 1/2 lb of salt in solution. Water containing a salt concentration of 2001 (10 − t)^2 (sin(t) +1) lb per gallon flows into the tank at a rate of 1 gal/ min, and the mixture is allowed to flow out of the tank at a rate of 2 gal/ min. The mixture is kept uniform by stirring. Let Q(t) (in lb) be the amount of salt in the tank after time t (in min).
(i) How long (in min) will it take for the tank to become empty?
Answer: t = 10 min.
(ii) Write the initial value problem for Q(t) (before the tank is empty) and solve it.
Answer:
Q(t) =
(10 − t)^2 (t − cos(t) + 2).
- Solve the differential equation
(2xy + y^3 )dx + (x^2 + 3xy^2 − 2 y)dy = 0.
Answer:
x^2 y + xy^3 − y^2 = C.
- Find the general solution of
dx¯ dt
¯x.
Answer:
x ¯ = c 1
cos(t) + sin(t) cos(t)
e^5 t^ + c 2
cos(t) − sin(t) − sin(t)
e^5 t.
(ii) Draw the phase portrait of the system. What is the origin called in this case? Is the origin stable, unstable or semi-stable?
Answer: The origin is an unstable clockwise spiral.
- Use the method of undetermined coefficients to solve the initial value problem
y′′^ − 4 y′^ +13y = (4t−4) cos(3t)+(12t−6) sin(3t), y(0) = 1, y′(0) = 1.
Answer:
y(t) = t cos(3t) + e^2 t^ cos(3t) −
e^2 t^ sin(3t).
- Find the solution of the given initial value problem.
y′′^ + 4y = g(t), y(0) = 0, y′(0) = 0,
where
g(t) =
1 , t < π, sin(t), t ≥ π.
The sine Fourier series of f (x) = 2x + 1 over [0, 1] is
∑^ ∞
n=
nπ
sin(2nπx) +
∑^ ∞
n=
[2n − 1]π
sin((2n − 1)πx).
You could have written the cosine Fourier series as well.
- Solve the heat conduction problem:
uxx = 4ut, 0 < x < π, t > 0;
u(0, t) = 10, ux(π, t) = 0, t > 0;
u(x, 0) = sin(3x/2) + 10, 0 < x < π.
Hint: Here the left end of the bar is kept at fixed temperature 10, while the right end is insulated. There is no ”ready formula” for this case, you have to work from scratch.
Answer:
u(x, t) = e−^9 t/^16 sin(
3 x 2
- Solve the wave problem:
uxx = utt, 0 < x < π, t > 0;
u(0, t) = 0, u(π, t) = 0, t ≥ 0;
u(x, 0) = sin(4x) + 2 sin(6x), ut(x, 0) = sin(10x), 0 ≤ x ≤ π.
Answer:
u(x, t) = sin(4x) cos(4t) + 2 sin(6x) cos(6t) +
sin(10x) sin(10t).
