Differential Equations and Linear Algebra
Maria Isabel Bueno Cachadina South Hall 6524 [email protected] Ofce hours: Mondays and Thursdays from 4:00-5:30 pm
Differential Equations and Linear Algebra
Syllabus
Textbook: Differential Equations and Linear
Josh Ahn America is in the Heart An old worn down building is shown. The drab grey paint is peeling an there is a sign painted on the building which reads Caf. Camera follows Carlos who is dressed in shabby yet somewhat decent looking trousers and a white
Dierential Equations and Linear Algebra
Math 3C, Summer Session B 2011
Instructor: Liang Chang
August 14, 2011
Answer the following 5 questions. Calculators are not allowed. The use
of books or notes of any kind is not allowed. Show all your work for full
1. True/False Questions. (enclose the answer with a circle)
1.
2.
3.
4.
5.
2.
Given any two n n matrices, A and B, AB = 0 means at
least one of the two matrices is equal to 0:
Given any two n n matrices, A and B, |AB| = |BA|:
Given an n n matrix A, Ax = b
Exercise
A ball is dropped at rest from a certain height. Its velocity after t seconds due to the earths gravitational eld is given by v (t ) = 32.2t ft/sec,
(when air resistance is ignored). How far has the ball fallen after t seconds?
M.I. Bueno
Differe
Most differential equations have an innite number of solutions. In general, all solutions of a rst-order DE form a family of solutions expressed with a single parameter C . Such a family is called the general solution. A member of the family that results
Leibniz Integral rule-Necessary for homework
The Leibniz integral rule gives a formula for differentiation of a denite integral whose limits are functions of the differential variable, z
b (z ) b (z )
f (x , z )dx =
a (z ) a (z )
db da f dx + f (b(z ), z
We say that two curves intersecting on a point (t , y ) are orthogonal at that point if the tangent lines to both curves at the intersection point are orthogonal.
M.I. Bueno
Differential Equations and Linear Algebra
Consider two families F1 and F2 . We sa
Eulers method
For the initial-value problem y = f (t , y ), use the formulas tn+1 = tn + h, yn+1 = yn + hf (tn , yn ), to compute iteratively the points (t1 , y1 ), ., (tn , yn ), using step size h. The piecewise-linear function connecting these points is
When does a differential equation have at least one solution? EXISTENCE. When does the model produce only one solution for a particular set of initial conditions? UNIQUENESS.
M.I. Bueno
Differential Equations and Linear Algebra
Consider the differential e
Euler-Lagrange method for solving linear rst-order differential equations
To solve a linear DE y + a(t )y = f (t ), where a and f are continuous on a domain I , take the following steps: 1. Solve y + a(t )y = 0 by separation of variables to obtain yh = Ce
Growth and Decay Phenomena
Some standard rst-order differential equation models: The exponential growth: The population P is growing at a rate proportional to the population at any time t dP = kP , dt k > 0.
Thomas Malthus used this model to estimate the
Nonlinear models: Logistic equation
The exponential growth model is linear and predicts exponential growth of a population. This kind of growth may occur in the initial stages, but it cannot continue indenitely. For long-range predictions, we need models
Math 3C
Practice Final Problems
Solutions
Prepared by Vince Zaccone
For Campus Learning
Assistance Services at UCSB
1a) (find a general solution)
y + 2y = 7
First we solve the homogeneous equation:
y + 2 y = 0
separate. int egrate.
yh = Ce 2t
Now find a p
Robertus Eric Prelab 4
1. -500 mV
2. -50 Volt, measured as 50 Volt
3. 510 mV
4.
5. VA=VB=0
No currents drawn
The objective for the lab is to build opamp circuit with inverting and non-inverting
configuration. Circuit inverting will be used as input for no