MAT 275 TEST 1 PRACTICE
1. Determine if each of the following equations is separable (Yes or No), and /or linear (Yes or No).
Record your answer in the following table. Do not attempt to solve the equations.
Equation
Separable
Linear
𝑦′=𝑡 + 1
𝑦𝑡
𝑦′=𝑦𝑡
𝑡 + 1
𝑦′=cos(𝑡𝑦)
𝑦′−𝑡𝑦 = 𝑡3
2. Consider the ODE 𝑦′= 3𝑦4+ 3𝑦3− 6𝑦2
a) Determine all the equilibrium (constant) solutions.
b) For what values of 𝑦 is 𝑦 increasing?
3. Which of the following differential equations best
represents the slope field at the right?
a) 𝑦′= 𝑥 + 𝑦
b) 𝑦′= 𝑥 − 𝑦
c) 𝑦′= 𝑦 − 𝑥
d) 𝑦′=𝑥𝑦
e) 𝑦′=𝑥
𝑦
4. Consider the differential equation: 𝑥2𝑦′′ − 3𝑥𝑦′+ 5𝑦 = 2𝑥2ln(𝑥)
(a) What is the order of the differential equation?
(b) Is the differential equation linear or nonlinear?
5. Consider the differential equation: −4𝑥 + 𝑦2+ 2𝑥𝑦𝑦′= 0.
(a) What is the order of the differential equation?
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6. Determine the value(s) of the constant 𝑟 so that 𝑦 = 𝑒𝑟𝑡 is a solution to 𝑦′′ − 𝑦′− 6𝑦 = 0.
7. The general solution of the differential equation 𝑥𝑑𝑦 = 𝑦𝑑𝑥 is a family of (determine the correct choice)
a) circles b) parabolas c) hyperbolas d) lines passing through the origin
8. Use separation of variables to find the solution of the following Initial Value Problems.
Write your answer in explicit form and simplify as much as possible. For each IVP determine the
interval in which the solution is defined.
a) 𝑦′=2𝑥
𝑦+1 ,𝑦(1)= −2
b) 𝑥𝑦2+ 3𝑦2− 𝑥2𝑦′= 0,𝑦(1)= 3
c) 𝑦′=2𝑡
𝑡2𝑦+𝑦 ,𝑦(1)= 2
d) 𝑥2
𝑦2−3 𝑑𝑦
𝑑𝑥 =1
2𝑦 ,𝑦(1)= 2
e) 𝑑𝑦
𝑑𝑥 =20𝑦𝑥4,𝑦(0)= 4
9. A tank initially contains 60 gal of pure water. Brine containing 1 lb of salt per gallon enters the tank at
2 gal/min, and the (perfectly mixed) solution leaves the tank at 3 gal/min; thus the tank is empty after
exactly 1 hour. Let y(t) be the amount of salt in the tank after t minutes.
(a) Write an Initial Value Problem for the amount of salt in the tank at any time t (< 60).
(b) Solve the IVP in part (a) to find the amount of salt in the tank at any time t (< 60).
(c) Determine the amount of salt when the tank is half empty.
10. A completely filled 20 gallon tank originally contains 10 pounds of salt dissolved in water.
Pure water enters the tank at the rate of 5 gallons/minute, and the well-stirred mixture leaves the tank
at the same rate. Find the amount of salt in the tank at any time t .
11. A ball with mass 0.2 kg is thrown upwards with initial velocity 26 meters per second. We assume that the
forces acting on the body are the force of gravity and a retarding force of air resistance with direction opposite to
the direction of motion and with magnitude 𝑐|𝑣(𝑡)| where 𝑐 = 0.1𝑘𝑔
𝑠 and 𝑣(𝑡) is the velocity of the ball at time 𝑡.
The gravitational constant is 𝑔 = 9.8𝑚
𝑠2 .
(a) Write and solve the differential equation for the velocity, 𝑣(𝑡).
12. A field mouse population satisfies the Initial Value Problem: 𝑑𝑝
𝑑𝑡 = 0.5𝑝 − 400, 𝑝(0)=600.
(a) Find the time at which the population becomes extinct.
(b) Find the time at which the population becomes extinct if the initial condition is 𝑝(0)=800.
13. Newton's law of cooling is 𝑢′= −𝑘(𝑢 − 𝑇) where u(t) is the temperature of an object, t is in hours, T is a
constant ambient temperature, and k is a positive constant. Suppose a building loses heat in accordance with
Newton's law of cooling. Suppose that the rate constant k has the value 0.13ℎ𝑟−1. Assume that the interior
temperature of the building is 76°F when the heating system fails and the external temperature is T=10°F.
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(a) How long will it take for the interior temperature to fall to 32°F?
(b) What happens to the temperature u(t) as 𝑡 → ∞?
14. Suppose P(t) denotes the size of an animal population at time t and its growth is described by the differential
equation 𝑑𝑃
𝑑𝑡 = 0.001𝑃(1000 − 𝑃). Determine the value of P at which the population is growing fastest.
15. Solve the following Initial Value Problems using the method of integrating factor.
a) 𝑦′=2
𝑡𝑦 + 6𝑡4,𝑦(1)= −3
b) 𝑦′− 2𝑦 = 2𝑒5𝑡 + 5𝑒2𝑡,𝑦(0)= −3
c) 𝑡𝑦′= −sin(𝑡)
𝑡− 2𝑦,𝑦(𝜋)= 1
d) (𝑡 + 1)𝑦′− 2𝑦 = 2𝑡,𝑦(0)= 4
e) 𝑑𝑦
𝑑𝑡 + 0.2𝑡𝑦 = 5𝑡,𝑦(0)= 6
16. Determine (without solving the problem) the maximal interval in which the solution of the given initial value
problem is guaranteed to exist: 𝑡𝑦′+ tan(𝑡)𝑦 = sin(𝑡),𝑦(𝜋)= 6.
17. a) Verify that both 𝑦1= 2𝑡 − 1 and 𝑦2= 𝑡2 are solutions to 𝑦′= 2(𝑡 − √𝑡2− 𝑦) .
In which intervals in t are the solutions valid?
b) Does the existence of two solutions of the given problem contradict any known theorem about
existence and uniqueness of solutions to Differential Equations?
18. Consider the following differential equations. Determine if the Existence and Uniqueness Theorem does or
does not guarantee existence and uniqueness of a solution of each of the following initial value problems.
I. 𝑑𝑦
𝑑𝑥 =√𝑥 − 𝑦,𝑦(2)= 2
II. 𝑑𝑦
𝑑𝑥 =√𝑥 − 𝑦,𝑦(2)= 1
III. 𝑦𝑑𝑦
𝑑𝑥 = 𝑥 − 1,𝑦(0)= 1
IV. 𝑦𝑑𝑦
𝑑𝑥 = 𝑥 − 1,𝑦(1)= 0
22. Use Euler's method and two steps with ∆𝑥 = 0.1 for the differential equation y' = y, with initial value
𝑦(0)= 1, to find the approximate value of 𝑦(0.2).
23. Use Euler's method and three steps with ∆𝑡 = 0.2 for the differential equation 𝑦′= 3𝑡 − 2𝑦2− 2, with
initial value 𝑦(1)= 1, to find the approximate value of 𝑦(1.6).
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