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Math 287 Sample Questions (from previous exams)
Chapter 1
- By differentiation and substitution determine which of the following families of functions is a solution to the differential
equation x
y
dx
dy
- For both families show your work necessary to determine if the family is a solution and state
your conclusion for each family
(a) (b)
x y ce
−
12 2 y = 2 ( 12 − x )+ c ( 12 − x )
- Given your answer to problem (1), find a solution to the initial value problem x
y
dx
dy
2 ,^ y ( 2 )= 3
- Find the general solution to the differential equation 2 1
y
t y. Write your solution in explicit form to receive full credit
- To the right is the graph of f ( y ).
(a) Sketch a slope field for f ( y ) dx
dy
(b) Determine any equilibrium solution(s) of f ( y ) dx
dy = and the basin(s) of attraction for these solutions.
Problems 5 - 7 refer to the IVP t
y y −
4 y ( 1 )= 3. Give your answers to four decimal places
- (a) Use Euler’s Method with a step size of h=0.5 to approximate y(2). You must show your work to receive any credit.
Record the results of any intermediate calculations in the chart below
n t (^) n y (^) n f ( tn , yn )
(b) Draw the path that Euler’s method takes through the direction field of the equation
- When h = 0.2 h = 0.1approximate Euler’s Method gives the approximation y (5) = 7.5882 y (2) = 5.6770. Use
you answer to problem (5), this new Euler approximation for h = 0.1and Richardson’s Extrapolation to determine a third
approximation for y (2).
- Use the fourth order Runge-Kutta method with a step size of h^ =^4 to approximate y ( 5 ), a solution to the IVP
t
y y −
4 y ( 1 )= 3. You must show your work to receive any credit.
- graph the isoclines for the differential equation 2 1 for the slopes
2 y ′ = t − y + − 2 , 0 ,and 2
- On the graph you drew in problem 8 now draw the direction field of the differential equation
- To the right is the slope field for the differential
equation f ( x , y ) dx
dy
(a) Sketch a graph of the solution to the IVP consisting of the given differential equation
and the initial condition y ( − 1 )=− 2
(b) Is the differential equation an autonomous equation? Explain your reasoning
- The following differential equation is Euler-Homogeneous. Using the change of variables , where v is an unknown
function of t , transform the following differential equation into a separable differential equation of the form
y = vt
v ′^ = f v t ( , )
(You do NOT have solve the differential equation) 2 2 dy y t
dt yt
12 Picard’s Existance Uniqueness Theorem guarantees that each of the following IVP’s will have a solution. However, the
theorem only guarantees that one of the IVP’s will have a unique solution. Which IVP is guaranteed to have a unique solution?
Explain why your choice does have a unique solution and the other IVP does not.
(a)
2 3 y = y , y (1) = 0 (b)^
2 3 y = y , y (1) = 1
- Match the differential equation with its slope field. (see problems 13 - 18 in section 1.2 homework)
- Match the differential equation with its direction field. NOTE: One DE and one direction field will NOT have a match.
Write NONE on the direction field without a match and next to the DE without a match
(A) y ' = (3 − y )( y +1)
(B)
y ' y t
(C) '
y y y t
(D) y '= y − t
- Write a first-order differential equation that expresses the following situation: The rate of increase of the grade point
average of a student is directly proportional to the number of study hours per week and inversely proportional to the
square root of the number of hours
G N
H spent watching television.
- (a) Use Euler’s method with a step size of 0.5 to approximate the solution at t = 4 , rounded to 4 decimal places, to the initial
value problem y '= t + y , y (2) = 6. Record the results of any intermediate calculations in the table below
n
The true value for y (4)(rounded to four decimal places) is 61.
(b) Determine the absolute error for your approximation in part (a)
(c) Estimate the approximation Euler’s method would give if we were to use a step size of 0.05, that is 1/ th of the step size
you used in part (a), to approximate. (HINT: To make this estimate you will need the true value, your answer to part (b),
the order of Euler’s method.) Do
y (4)
NOT try to answer this question by actually using Euler’s method with a step size of 0.05)
- I used the same order 2, that is an , numerical method twice to approximate the solution at to the IVP . The results are in the table below
2 Ο( h ) t = 3
2 y ' = y − y + t , y (0) = 1
Step Size (^) Approximation of y (3)
(a) Which approximation, 4.85416 or 5.09655 should be closer to the true value of (^) y (3)?
(b) Use the information I have provided and Richardson’s Extrapolation to produce a third, and even more accurate
approximation of y (3)
- In this problem you will be applying Picard’s theorem to the IVP. 3 y ' = t y − 2, y (4) = 2
Chapter 2
1. Solve the following equation using the method of Variation of Parameters 1
2
dy t
2. Suppose y 1 (^) ( t )and y (^) 2 ( t )are both solutions to the differential equation a y t dt
dy a (^) 1 + 0 = 2. Prove or disprove that
is also a solution to the differential equation. You must show your work and your reasoning must be clear to
receive credit.
y 1 (^) ( t )+ y 2 ( t )
3. This problem concerns the differential equation ty ty y
dt
dy
(a) Use the change of variable to transform the equation into a linear differential equation in the variables t and
z y = e z
(b) Solve the linear equation from part (a) using the Integration Factor method (half credit for this part if you use another
method)
(c) Now find the solution to the Initial Value Problem ty ty y dt
dy
4. Let y ( t )be the number of trout in an imaginary lake t years after today.
(a) Develop the differential equation (BUT DO NOT SOLVE THE DE) that models the trout population according to each of the following assumptions: Model 1: The population of trout grows with a constant relative growth rate of 0. Model 2: The population grows as in Model 1 for small values of but the relative growth rate slows down as
approaches 10,000, the maximum number of trout the lake’s ecosystem can support.
y y
Model 3: The population grows as in Model 2 and fishermen catch trout at a constant annual rate of 200 trout per year. (b) Draw the phase line for model (3) (c) Find the maximum growth rate for the trout population using model 3
(d) Determine the long term trout population (i.e. the steady state population) using the assumptions in Model 3 with the
following initial conditions
y ( 0 )= 1000 y ( 0 )= 5000 y ( 0 )= 9000
(e) (extra credit) Let be the parameter equal to the number of fish caught per year in Model 3. Varying this parameter will
change the solutions to the Model. Determine the bifurcation point for parameter.
F
5. A thermometer that initially reads is moved to a room where the air temperature is.
D 50
D 70 (a) Assuming the temperature in the room does not change, write down the Initial Value Problem (BUT DO NOT SOLVE
THE IVP) whose solution will give the temperature T of the thermometer after it has been in the room for t minutes.
(b) Now, suppose the air temperature in the room is not constant but instead is rising at a rate of per minute. Write
down the Initial Value Problem (BUT DO NOT SOLVE THE IVP) whose solution will give the temperature T of the thermometer after it has been in the room for t minutes.
D 2
(c) Using the same assumptions as in part (b) assume further that the temperature of the thermometer is after 10 minutes. Determine the EXPLICIT form of the function that gives the temperature of the thermometer as a function of time spent in the room. (That is, find the explicit solution to the IVP)
D 65
- A 100 gallon tank initially contains 60 lb of salt dissolved in 60 gallons of water. A salt solution with a concentration of 2
lb/gal flows into the tank at a rate of 4 gal/min, and the well-stirred mixture flows out of the tank at a rate of 3 gal/min.
(a) Write the IVP for x ( ) t , the amount of salt in the tank after t minutes.
(b) Solve the IVP for the amount of salt in the tank and determine the number of pounds of salt in the tank when the tank is full. (b) Graph the nullclines on the phase portrait given
- Match the differential equation with its direction field
(A) y ′ = ( y − 2)( y + 2 )
(B) ( )
1 2 4 y ′ = ( y − 2) ( y + 2 )
(C) ( )
1 2 4 y ′ = ( y − 2)( y +2)
(D) ( )
1 2 2 y (^) 6 ( y 2) ( y 2)
- (a) Find the general solution to the homogeneous differential equation (^) ty ′ (^) = 3 y
(b) Use the technique of variation of parameters to find a particular solution to the nonhomogeneous equation 5 ty ′ = 3 y + t sin t
(c) Determine the general solution to ty
5 ′ = 3 y + t sin t
- A 30 year old woman accepts an engineering position with a starting salary of 40,000 per year.
(a) Her salary increases at the continuous rate of 5%. Write down the IVP for her salary and the solution to this IVP. (b) She continuously deposits 12% of her salary into a retirement account which earns interest at a continuous rate of 6%. (i) Write down the IVP for the amount of money in her retirement account after t years (ii) Solve the IVP for the amount of money in her retirement account. (iii) Determine how much money will be in her account when she is 65 years old.
- The following differential equation models a population living on an island that is being hunted at a constant annual rate
dy y y dt
(a) What are the steady state, that is equilibrium, populations?
(b) Solve the differential equation
