math 306d - university at buffalo

MATH 306D

Final Exam

May 11, 2020

NAME (please print legibly):

Your University ID Number:

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-1

f(t) F (s) eat 1s−a

a f(t) + b g(t) aF (s) + bG(s) tneat n!(s−a)n+1

f ′(t) sF (s)− f(0) cos kt ss2+k2

f ′′(t) s2F (s)− sf(0)− f ′(0) sin kt ks2+k2

f (n)(t) snF (s)− s(n−1)f(0)− · · · − f (n−1)(0) cosh kt ss2−k2∫ t

0f(τ)dτ F (s)

ssinh kt k

s2−k2

e−atf(t) F (s+ a) eat cos kt s−a(s−a)2+k2

u(t− a)f(t− a) e−asF (s) eat sin kt k(s−a)2+k2∫ t

0f(τ)g(t− τ)dτ F (s)G(s) 1

2k3(sin kt− kt cos kt) 1

(s2+k2)2

t f(t) −F ′(s) t2k

sin kt s(s2+k2)2

tn f(t) (−1)nF (n)(s) 12k

(sin kt+ kt cos kt) s2

(s2+k2)2

f(t)t

∫∞sF (σ)dσ u(t− a) e−as

s

f(t), period p 11−e−ps

∫ p0e−stf(t)dt δ(t− a) e−as

1 1s

tn n!sn+1

t 1s2

1√πt

1√s

0

Questions 1-5 (part A) are on material since Test 3.

1. (10 points)

1.1: Find the function whose Laplace transform is s−2s2+4

1.2: Use the result from 1.1, to find the function whose Laplace transform is (s+3)−2(s+3)2+4

Hint: L−1{F (s+ a)} = e−atf(t) = e−at where f(t) = L−1{F (s)}.

1.3: Use the result from 1.2, to find the function whose Laplace transform is e−s((s+3)−2)(s+3)2+4

Hint: L−1{e−asF (s)} = u(t− a)f(t− a) where f(t) = L−1{F (s)}.

1

2. (10 points) An object whose mass is 1 kg is attached to the bottom of a spring,

stretching the spring by 52

meters. Let k be the spring constant, as calculated assuming that

the acceleration due to gravity g is equal to 10 meters/sec2. Recall k∆y = mg.

An attached damper has damping coefficient c = 4 Newton/(meter/sec).

An external force equal to e−2 t Newton is applied, acting downward.

Let y(t) be the displacement of the object below its equilibrium position. The object is

released at t = 0, from initial position y(0) = 0 and with no initial velocity: y′(0) = 0.

2.1: Write down differential equation for y(t).

2.2: Find the formula for Y (s), the Laplace transform of y(t).

2.3: Find y(t) = L−1(Y (s)).

2

3. (10 points) Use Laplace transforms to solve the initial value problem

y′′ (t) + y (t) = u (t− 4) , y (0) = 0, y′ (0) = 0.

Show all your work. (Note: Find Y (s), the Laplace transform of y(t). Partial fraction

decomposition of Y (s) will be needed before using the table.)

3

4. (10 points) Consider the differential equation

y′′ + y′ − 2y = 0

4.1: Find the recurrence relation for the coefficients of the power series y(x) =∑∞

n=0 cnxn

4.2: Find the coefficients c0, c1, c2, c3 and c4 of the power series solution such that y(0) = 1

and y′(0) = 1.

4.3: Guess the formula for the coefficients cn for n = 0, 1, 2, . . . .

4

5. (10 points)

5.1: Find the values of r for which the DE

4x2y′′(x) + 2xy′(x)− xy(x) = 0

has Frobenius series solutions y(x) =∑∞

n=0 cnxn+r, x > 0, c0 6= 0.

Hint: The values of r are the solutions of the indicial equation,

obtained from the term of the series for DE for the lowest power of x.

5.2: Find the recurrence relation for the larger of the roots r, then assume c0 = 1 and solve

for the coefficients c1, c2 and c3.

5

In questions 6-10 (part B), DO NOT use Laplace transforms or series solutions.

6. (10 points)

6.1 (8pts) Find the general solution y(x) for the first-order linear differential equation:

xd

dxy (x) + 2 y (x) = 4 x−1 + 6, for x > 0

6.2: (2pts) Find the solution of the above equation which satisfies y(1) = 12.

6

7. (10 points)

7.1: (3pts) Sketch the phase line for dxdt

= x(x− 1) on a vertical scale such as on the

right, taking the x-positive direction, as “up”.

Show directions of travel (up/down) by vertical arrows.

Show the equilibrium points as dots, and their stability with letters S or U.

7.2: (7pts) Solve by separation of variables: dxdt

= x(x− 1), x(0) = 2.

7

8. (10 points) (Recall: In part B, do NOT use Laplace transforms)

8.1: Consider the DE

y′′ − 3y′ + 2y = et .

Find a particular solution y = yp by the method of undetermined coefficients.

8.2: Solve the initial value problem y′′ − 3y′ + 2y = et, y(0) = 0, y′(0) = 0.

8

9. (10 points)

9.1 Find the general solution of x′ = Ax with A =

[1 1

4 1

].

9.2: Sketch the phase portrait for the system from above.

9

10. (10 points) Consider the nonlinear system

{x′(t) = f(x, y) ≡ x(4− x− y)

y′(t) = g(x, y) ≡ y(x− y − 2)

10.1: Of x and y, which is predator and which is prey?

10.2: Find the critical point (x0, y0) with x > 0, y > 0.

10.3: Compute the eigenvalues of the Jacobian matrix

(fx fy

gx gy

)at (x0, y0).

10.4: Sketch the phase portrait for the nonlinear system above.

Include behavior near the critical point (x0, y0) and include arrows representing di-

rection of travel for representative trajectories that belong to the y−axis and to the

x−axis. Note that the system has FOUR critical points.

10