Math 225
Fall 2005
Intro Differential Equations
Final Review
Name:
Page 1
This is a 75 min exam. Please make sure you put your name on the top right hand corner of
each sheet. Textbooks, open notes or calculators are NOT allowed. Remember the Honors
Code will be enforced!
Problem 1: Improved Euler’s Method TBD POINTS
Work!
dy
dx
Show all
= y y(0) = 1
a) Use the improved Euler’s Method to approximate the solution of this IVP (namely ex ) at
the point x=1. Assume h=.10
b) e1 = 2.718281828. So what is
error
?
h
c) What order is the improved Euler’s method? and hence what do you expect to happen
to the ratio error
as h −→ 0?
h
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Math 225
Fall 2005
Intro Differential Equations
Final Review
Problem 1: Euler’s Method Workspace
2
Name:
Page 2
Math 225
Fall 2005
Intro Differential Equations
Final Review
Name:
Page 3
Problem 2: Auxiliary equations and Implication of Roots TBD
POINTS Show all Work!
y 00 + αy 0 + y = e−t y(0) = 1 and y 0 (0) = 0
a) Write down the auxiliary equation for the corresponding homogeneous system of the IVP
above.
b) Determine the values of α so the auxiliary equation has:
Case1: A double real root
Case2: Two real roots
Case3: Complex conjugate pair roots
c) What are the roots for each case above.
Please leave α as a parameter in your answers but describe its constraint in each case.
d) Write down the homogeneous solution for each case above.
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Math 225
Fall 2005
Intro Differential Equations
Final Review
Name:
Page 4
Problem 2: Auxiliary equations and Implication of Roots Workspace
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Math 225
Fall 2005
Intro Differential Equations
Final Review
Name:
Page 5
Problem 3: Method of Undetermined Coefficients TBD POINTS Show
all Work!
y 00 + αy 0 + y = e−t y(0) = 1, y’(0)=0
a) Determine the particular solution to the ODE above if
α = 0:
α = 2:
α = 3:
using the Method of Undetermined Coefficients.
b) What is the general solution to the ODE for α = 2.
c) Determine the solution to the IVP for α = 2. (apply initial conditions)
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Math 225
Fall 2005
Intro Differential Equations
Final Review
Name:
Page 6
Problem 3: Method of Undetermined Coefficients Workspace
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Math 225
Fall 2005
Intro Differential Equations
Final Review
Name:
Page 7
Problem 4: Mass-Spring Oscillator and Phase Plane TBD POINTS Show
all Work!
a) Draw a sketch of a Mass-Spring Oscillator with a single mass (1kg) and single spring
(k=1) attaching the mass to a wall. Assign y=0 to the rest position of the mass.
b) Derive the second-order ODE equation of motion of the mass for the case when there is
dampening called b = 4 and NO forcing function.
c) Using the assignments x1 = y and x2 = y 0 decompose this single second-order ODE into
two first order ODEs.
d) Find the critical points for this system.
e) Put the system into matrix normal form.
f) Solve the system using eigenvalues and eigenvectors for the homogeneous solution.
y
y0
!
so we get both the position function and the velocity function at once.
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Math 225
Fall 2005
Intro Differential Equations
Final Review
Name:
Page 8
Problem 4: Mass-Spring Oscillator and Phase Plane Workspace
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Math 225
Fall 2005
Intro Differential Equations
Final Review
Name:
Page 9
Problem 5: Variation of Parameters TBD POINTS Show all Work!
2y 00 − 2y − 4y = 2e3t
Use variation of parameters to find the particular solution for this non-homogeneous ODE.
Show all steps of the method.
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Math 225
Fall 2005
Intro Differential Equations
Final Review
Name:
Page 10
Large Problem 6: Solving IVP using Laplace transform TBD POINTS Show
all Work!
Solve the IVP using Laplace transform methods.
y 00 − 7y 0 + 10y = 9cos(t) + 7sin(t) y(0) = 5 and y 0 (0) = −4
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Math 225
Fall 2005
Intro Differential Equations
Final Review
Name:
Page 11
Large Problem 7: Definition and properties of Laplace transform
TBD POINTS Show all Work!
a): What is the definition of the Laplace transform?
b): Using the definition, derive the Laplace transform of the function t.
c): Express the following discontinuous function with unit step functions.
t,
0<t<1
2,
1<t<2
f (t) =
(t − 2)2 ,
t≥2
d): Using the properties of the unit step, what is the Laplace transform of f(t).
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Math 225
Fall 2005
Intro Differential Equations
Final Review
Name:
Page 12
Problem 8: Solve using separation of variables. TBD POINTS Show
all Work!
Solve the following IVP using separation of variables.
y 0 = x3 (1 − y) y(0) = 3
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Math 225
Fall 2005
Intro Differential Equations
Final Review
Name:
Page 13
Problem 9: Solve using Integrating Factors TBD POINTS Show
all Work!
Solve the following IVP using Integrating factors
dy
dx
+ 4y − e−x = 0 y(0) =
4
3
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Math 225
Fall 2005
Intro Differential Equations
Final Review
Name:
Page 14
Problem 10: Identify and Solve Exact Equations TBD POINTS Show
all Work!
Show the following problem is exact and then solve it.
( yt )dy + (1 + ln(y))dt = 0
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Math 225
Fall 2005
Intro Differential Equations
Final Review
Name:
Problem 11: True or False TBD POINTS Show all Work!
1. All theorem statements (hypothesis and conclusions) are fair game.
2. Comparing the process of solving IVPs with Laplace and without.
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