MATH 2410 PRACTICE PROBLEMS FOR FINAL EXAM Date and place: Saturday, December 16, 2017. Section 001: 3:30-5:30 pm at MONT 225 Section 012: 8:00-10:00am at WSRH 112. Material covered: Lectures, quizzes, worksheets, homework, and practice exams/problems. Policies: No calculators will be allowed. Format of the Exam: The format of the final exam will be similar to the practice test below. The total is 105 points, but you cannot earn more than 100 points. The types of problems will be similar to the practice test and the additional practice problems below. The distribution of points roughly breaks down as follows. Before Midterm 1 (23 points) Section 1.1 1.9, 2.1 2.2 Between two midterms(23 points) Section 2.3 2.6, 3.1 3.5, 3.7 and 5.1(no equal eigenvalues case) Second order differential equations (29 points) Section 3.6, 4.1 4.3 Laplace transform (30 points) 6.1 6.3 Be prepared to have a full two-hours test. Practice Exam: Short questions (3 points 8 = 24 points) Question A. Set up the differential equation for the following word problem. A 30-gallon tank initially contains 15 gallons of salt water containing 6 pounds of salt. Suppose salt water containing 1 pound of salt per gallon is pumped into the top of the tank at the rate of 2 gallons per minute, while a well-mixed solution leaves the bottom of the tank at a rate of 1 gallon per minute. Question B. Give the system of differential equations that models the following ecological system. On a small island, there are two species: rabbits and feral cats. Let R(t) and C(t) be the population of the two species (in thousands). Suppose that both rabbits population R(t) and Cats population C(t) satisfy logistic model with capacity 2 and growth rate 3. Suppose furthermore that the cats will attack the rabbits causing the rabbit population to drop at the rate of 1 2RC. Suppose such attack will not help the population growth of the Cats. Question C. Consider the system of differential equations ( ) dy 1 α dt = Y. 2 3 For which value of the parameter α is the system a source? Question D. Consider the differential equation y = 2ty +2t. Suppose that we know that y 1 (t) = 1 and y 2 (t) = 1+e t2 are solutions to this differential equation (with initial values y 1 (0) = 1 and y 2 (0) = 2). Consider the solution y 0 (t) with initial value y3(0) = 3 2 2. What is lim t y3(t)? 2 1
dy dt Question E. What special properties do the slope field of differential equations of the type = f(t) have? Suppose that we have one solution curve. Can we get other solutions? Question F. Find the equilibrium solutions to the following system. { dx dt = 3x(2 x) dy dt = 2y(4 y)+4xy. Question G. Find the following inverse Laplace transform. L 1[ 2s+3 ]. s 2 +6s+13 Problem H. Find the following Laplace transform L [ u 2 (t)e a(t 2) ]. Problem 1. (9 points) Consider the differential equation dy dt = (y 2)2 ycosy. Sketch the phase line when y [ 5,5] and classify the equilibrium points in that range as sinks, sources, or nodes. Draw a rough sketch of the solution of with initial value y(0) = 3. Problem 2. (8 points) Solve the following initial value problem dy dt = (2t+1)y +e t2, y(0) = 0. Problem 3. (7 points) Find the general solution to the following linear system, and sketch its phase portrait. ( ) dy 1 1 dt = Y. 2 3 Problem 4. (7 points) Find the general solution to the following linear system, and sketch its phase portrait. ( ) dy 1 0 dt = Y. 2 3 Problem 5. (7 points) Solve the initial value problem y +4y +5y = e 2t, y(0) = y (0) = 0. Sketch the graph of the solution. Moreover, if we start with another initial value, what does the solution look like when t, and WHY? Problem 6. (8 points) Give the general solution to the following differential equation y +4y +3y = 10cost. Find the solution with initial value y 0 (0) = y 0 (0) = 0. Draw the graph to indicate both the particular solution and the solution to the initial value above. Discuss their long-term behavior. Problem 7. (11 points) Solve the initial value problem y +100y = cos9t, y(0) = y (0) = 0. 2
Determine the frequency of the beats and the frequency of the rapid oscillation. Sketch the solution to the given initial value. (In the actual final exam, if a problem like this is given, the following formula will be available.) cosα cosβ = 2sin α+β 2 sin α β 2. Problem 8. (6 points) Find the following inverse Laplace transform. L 1[ 7e 2s ]. (2s+3)(s 2) Problem 9. (8 points) Use Laplace transform to solve the following initial value problem. y = y +u 5 (t), y(0) = 3. Problem 10. (10 points) Use Laplace transform to solve the following initial value problem. Additional practice problems y +9y = 6, y(0) = 3, y (0) = 3. Problem I. Beth initially deposits $6,000 in a savings account that pays interest at the rate of 1% per year compounded continuously. She arranges for $20 per week to be deposited automatically into the account. Assume that weekly deposits are close enough to continuous deposits so that we can reasonably approximate her balance using a differential equation. Write an initial-value problem for her balance over time. Approximate Beth s balance after 4 years. Problem II. Consider the following eco-system. On a small island, there are two species: rabbits and foxes. Let R(t) and F(t) be the population of the two species (in thousands). Suppose that the rabbits population R(t) satisfies the logistic model with capacity 2 and growth rate 4. Suppose without rabbits, the fox population will decline at the rate 1 2. Moreover the foxes will eat rabbit at the rate of 5RF, and on average eating five rabbits will increase the population of fox by 1. (1) Use a system of differential equations to model the population dynamics of the two species. (2) Find the equilibrium points of the system. (3) Give the linearization at the equilibrium point(s) where neither R nor F is zero. (4) determine the type of the system at the equilibrium point considered in (3). Problem III. Consider the differential equation with parameter α dy dt = y2 4y +α. (1) Draw the phase line of the system when α = 3. Classify the equilibrium points as sinks, sources and nodes. Draw typical solutions with initial values in each intervals. (2) Draw the bifurcation diagram and compute the bifurcation value. Draw the phase lines for the system when α is slightly smaller than, slightly larger than, and equal to the bifurcation value. Problem IV. Solve the following initial value problem. dy dt = 1 e t +e t, y(0) = 1. y Problem V. Find the general solution to the following systems and sketch the phase portrait. ( dx ) ( )( ) dt 1 1 x (1) dy =. 4 1 y dt (2) dy ( ) 1 dt = 3 2 Y. 2 3 3
Problem VI. Solve the following initial value problem. y 6y +5y = 3e 2t, y(0) = y (0) = 0. Problem VII. Give the general solution to the following differential equation describing a damped oscillator with sinusoidal forcing. y +2y +2y = 5sint. Problem VIII. Find the following Laplace transform or inverse Laplace transform. (1) L 1[ e 4s ]. s 2 1 (2) L [ u 2 (t)e 3(t 2) sin(4(t 2)) ]. Problem IX. Use Laplace transform to solve the following initial value problems. (1) y = 5y +e t, y(0) = 1. (2) y +4y = sin3t, y(0) = 1, y (0) = 1. (3) y +6y +5y = 10, y(0) = 2, y (0) = 4. 4
Problem 8. We use partial fractions. Set 7 (2s+3)(s 2) = A s+ 3 + B s 2. 2 So we have 7 = 2A(s 2)+B(2s+3). Setting s = 2 gives 7 = B 7 and B = 1; setting s = 3 2 gives 7 = 2A ( 3 2 2) and A = 1. So L 1[ 7 ] = L 1 [ 1 ] + L 1 [ 1 ] = e 3t/2 +e 2t. (2s+3)(s 2) s 2 From this we deduce that L 1[ 7e 2s (2s+3)(s 2) Problem 9. y = y +u 5 (t), y(0) = 3. We apply Laplace transform to get We know that 1 s(s 1) = 1 s 1 1 s s+ 3 2 ] = u2 (t)e 3 2 (t 2) +u 2 (t)e 2(t 2). L[y ] = L[y]+L[u 5 (t)]. sl[y] 3 = L[y]+ e 5s s. (s 1)L[y] = 3+ e 5s s. L[y] = 3 s 1 + e 5s s(s 1). (by partial fractions). So we have L[y] = 3 s 1 + e 5s s 1 e 5s s. y(t) = L 1[ 3 ] + L 1 [ e 5s ] L 1 [ e 5s ] s 1 s 1 s = 3e t +u 5 (t)e t 5 u 5 (t) = 3e t +u 5 (t) ( e t 5 1 ). Problem 10. y +9y = 6, y(0) = 3, y (0) = 3. Take the Laplace transform of the given equation, we have L [ y ] +9L[y] = L[6]. Rewrite everything in terms of L[y]: ( s 2 L[y] sy(0) y (0) ) +9L[y] = 6 s. Plugging in the initial values, we have ( s 2 L[y] 3s 3 ) +9L[y] = 6 s. (s 2 +9)L[y] = 6 s +3s+3. L[y] = 6+3s+3s2 s(s 2. +9) 11