18.03 Recitation 1
Modeling: natural growth
A certain African government is trying to come up with a good policy regarding
the hunting of oryx in a specic game preserve. They are using the following
model: the oryx population has a positive natural growth
18.03 Recitation 0
We review separation of variables, a technique from one-variable calculus
(18.01) used to solve differential equations of the form
.2) = h(y)g(t)-
The main idea is to use differential (Leibniz) notation and rewrite the differ-
ential eq
18.03 Practice Solutions Final Exam, Spring 2015
(Taken from Fall 2013 Mainstream 18.03 Practice Final)
Problem 1.
(a)
Euler with n = 1 and x = 0.1 from (x0 , y0 ) = (0, 1): Slope m = y
(x0 ,y0 )
= 3;
y1 = y0 + m x = 1 + (3)(0.1) = 0.7. So y(0.1) 0.7 .
(b
18.03 Practice Questions Final Exam, Spring 2015
These are taken from the Spring 2014 mainstream practice exam. Some of the language
will be slightly different from what we used in our class. It will be good practice to try to
translate it.
Sometimes in t
18.03 Practice Questions Final Exam, Spring 2015
(Taken from Fall 2013 Mainstream 18.03 Practice Final)
Coecients of a Fourier series of period 2L
a0
If f (t) =
+
2
an =
1
L
an cos
n=1
nt
L
L
nt
L
f (t) cos
L
+
nt
, then
L
bn sin
n=1
dt
and
bn =
1
L
L
f (
18.03 Linear Algebra Practice Final Exam, Spring 2015
Problem 1
(a) Find the eigenvalues and eigenvectors of A =
3 13
.
2 1
(b) Find the eigenvalues and eigenvectors of A =
3 4
.
2 5
Problem 2
Suppose that the matrix B has eigenvalues 1 and 7, with eigenv
Review of entire semester, Spring 2015
Topic 1. Modeling; separable DEs
Problem 1. y = y(x) is a curve in the rst quadrant. The part of the tangent line in the
rst quadrant is bisected by the point of tangency. Find and solve the DE for this curve.
dy
y
y
18.03 Practice Solutions Final Exam, Spring 2015
Important: Not every topic is covered here. (E.g. population models with harvesting and
bifurcation diagrams.) When preparing for the nal be sure to look over old psets and exams.
y
1. (a) See picture at ri
Formula Sheet for Final Exam
Fourier Series for f (t), periodic with period 2L
An =
L
1
L
f (t) cos
L
nt dt,
L
Bn =
1
L
L
f (t) sin
L
nt dt
L
Period 2 square wave:
1 on < t < 0
1 on 0 < t <
sq(t) =
=
4
n odd
sin(nt)
n
Period 2 triangle wave:
tri(t) = |t|
18.03 Practice Final 2, Spring 2015 Solutions
This practice contains no linear algebra problems.
Problem 1.
(a) (12) Homogeneous equation: x + 3x + 4x = 0.
2 + 3r + 4 = 0 r = 3 9 16 = 3 7 i .
Characteristic equation: r
2
2
xh = c1 e3t/2 cos( 7 t) + c2 e3
18.03 Practice Questions Final Exam, Spring 2015
Important: Not every topic is covered here. (E.g. population models with harvesting and
bifurcation diagrams.) When preparing for the nal be sure to look over old psets and exams.
dy
y
= + 3 x:
dx
x
(a) Ske
Review of entire semester, Spring 2015
Topic 1. Modeling; separable DEs
Problem 1. y = y(x) is a curve in the rst quadrant. The part of the tangent line in the
rst quadrant is bisected by the point of tangency. Find and solve the DE for this curve.
Proble
18.03 Practice Solutions Exam 1, Spring 2015
dy
Problem 1 Separation of variables: cos y = sin xdx .
cos x
Integrating: ln(sec y + tan y) = ln | cos x| + C sec y + tan y = C1 | cos x|.
Lost solutions when cos y = 0 y = /2, y = 3/2 . . .
Problem 2
(a) In p
18.03 Practice Questions Exam 1, Spring 2015
(This will take considerably longer than 1 hour; the actual exam will be shorter.)
dy
Problem 1 Solve the following DE. (cos x )
+ (sin x cos y) = 0
dx
Problem 2
e(3+2i)x
3 + 2i
(where as usual Im denotes the i
18.03 Practice Exam 1b, Spring 2015 Solutions
Problem 1. (15 points)
dy
1
2
(a) (8) Separable:
= xdx = x2 /2 + C y = 2
.
2
y
y
x + 2C
Lost solutions when y 2 = 0 y(t) = 0.
(b) (7)
Linear rst order DE, unusual input: use variation of parameters formula.
yh
18.03 Practice Exam 1b, Spring 2015
8 problems, No books, notes or calculators.
Problem 1. (15 points)
dy
(a) (8) Solve the DE
= xy 2 .
dx
(b) (7) Give a denite integral solution to the following IVP y + t y = cos(t2 ); y(0) = 4.
Do not try to evaluate th
18.03 Topic 9: Applications: forced oscillations.
Author: Jeremy Orlo
Reading: SN FR; EP 2.6; 2.7 (up to Reactance & Impedance).
Applications (See 2.4 -reading and problems and also 2.7 for RLC circuits and
practical resonance)
Damped Forced Harmonic Osci
18.03 Topic 10: Direction elds, integral curves, existence of solutions.
Author: Jeremy Orlo
Reading: EP 1.3; SN G.1
Direction Fields
General rst order equation: y = f (x, y)
Examples: y = x y + 1, y = x2 + y 2 , .
Example: Consider the IVP y = x2 + y 2 ;
18.03 Topic 8: Applications, stability, variation of parameters.
Author: Jeremy Orlo
Reading: EP 2.4; SN S (pp.0-1); EP 2.5 (section on variation of parameters only).
Time invariance
Constant coecient have the property of time invariance. That is, if xp (
18.03 Topic 7: Inhomogeneous DEs: Operator and UC methods.
Author: Jeremy Orlo
Reading: SN O pp. 5-9.
Non-constant coecient DEs
Nice simple operator notation
General linear DE: y (n) + p1 (x)y (n1) + + pn (x)y = f (x)
Simpler form: write L = Dn + p1 (x)Dn
18.03 Topic 2: Linear Systems: input-response models.
Author: Jeremy Orlo
Reading: Class notes for this topic; SN IR1-4.
dy
+ p(t)y = q(t)
dt
Linear if y and y occur separately and only as rst powers.
y = ky
linear
y + y2 = t
Examples:
sin(t)
2
(y )2 + y
18.03 Topic 3: Input-response models continued.
Author: Jeremy Orlo
Reading: Class notes for this topic.
Input-response and systems
Like in theromodynamics the denition of system is somewhat arbitrary.
Example 1: Spring-mass:
m=mass, k=spring constant, x=
18.03 Topic 6: Operators, inhomogeneous DEs, exponential input theorem.
Author: Jeremy Orlo
Reading: SN O pp.1-5.
Linear Dierential Equations
General linear equations:
Inhomogeous standard form: y (n) + p1 (x)y (n1) + + pn (x)y = f (x)
Homogeous standard
18.03 Topic 4: Complex numbers and exponentials.
Author: Jeremy Orlo
Reading: SN C.
(Takes 1.5 classes)
Motivation: x2 + 1 = 0 has no real solutions.
We dene the complex solutions as 1 = i.
Were going to look at the algebra, geometry and, most important f
18.03 Topic 5: Linear DEs, CC homogeneous case.
Author: Jeremy Orlo
Reading: EP 2.3
(Takes 1.5 classes)
Second order constant coecient linear DEs:
x + bx + kx = f (t), where b, k are constants.
Second order: obvious.
Constant coecient: because coecients b
18.03 Topic 1: Introduction to DEs; modeling; separable equations.
Author: Jeremy Orlo
Reading: EP 1.1, 1.2 and 1.4 SN D.
Dierential equation: Di. eq. = equation with derivatives (Duh!)
Solution = any function that satises the DE
Examples: (DEs modeling p
18.03 HOMEWORK #1 PART B, DUE FRIDAY SEP 18, 12:55 PM 2015
This is Part B of your problem set. Part A, due at the same time, is at MITx.
You are encouraged to collaborate with other students in this class. But you must write
up your answers in your own wo
18.03 PROBLEM SET 5 PART B, DUE FRIDAY OCT. 23, 12:55 PM
This is Part B of your problem set. Part A, due at the same time, is at MITx.
You are encouraged to collaborate with other students in this class. But you must write
up your answers in your own word
18.03 PROBLEM SET 5 PART B, DUE FRIDAY OCT. 23, 12:55 PM
This is Part B of your problem set. Part A, due at the same time, is at MITx.
You are encouraged to collaborate with other students in this class. But you must write
up your answers in your own word