9.6: Matrix Exponential, Repeated Eigenvalues
x = Ax,
A:nn
(1)
Def.: If x1(t), . . . , xn(t) is a fundamental set of solutions (F.S.S.) of (1), then X (t) = [x1(t), . . . , xn (t)] (n n) is called a fundamental matrix (F.M.) for (1). General solution: (c
Mert Ozbayram
Assignment week10 due 08/22/2016 at 11:59pm EEST
math219
This problem set covers material from Sections 7.1 and 7.2.
Laplace transforms of basic functions.
Inverse Laplace transforms of basic functions.
Using Laplace transforms to solve 2
Mert Ozbayram
Assignment week8 due 08/22/2016 at 11:59pm EEST
math219
This problem set covers basic material from section 7.9 and Chapters 3 and 4.
Variation of parameters for nonhomogeneous systems (2x2).
Converting a higher order linear differential e
Mert Ozbayram
Assignment week9 due 08/22/2016 at 11:59pm EEST
math219
This problem set covers advanced material from Chapters 3 and 4.
Solving higher order linear differential equations.
Variation of parameters for higher order linear.
Undetermined coe
Mert Ozbayram
Assignment week11 due 08/23/2016 at 11:59pm EEST
math219
This problem set covers material from Sections 7.3 7.6.
Heaviside step functions and their Laplace transforms.
Solving initial value problems involving Heaviside step functions and D
MATH 219
Fall 2015
Lecture 5
ur Kisisel
Lecture notes by Ozg
Content: Exact equations and integrating factors (section 2.6).
Suggested Problems:
2.6: 1, 2, 7, 12, 14, 16, 19, 22, 32
1
Exact equations
Let us consider the case of an arbitrary first order O
MATH 219
Fall 2016
Lecture 2
ur Kisisel
Lecture notes by Ozg
Content: Linear equations; method of integrating factors. (section 2.1).
Suggested Problems:
2.1: 3, 6, 11, 15, 30, 33, 34
1
Linear Equations, Integrating Factors
In this section we will study
Mert Ozbayram
Assignment week12 due 08/23/2016 at 11:59pm EEST
math219
This problem set covers material from Chapter 10.
Boundary value problems.
Eigenfunctions.
Separation of variables.
Fourier series.
Heat equation.
Correct Answers:
1. (1 point) Solve t
MATH 219
Fall 2016
Lecture 3
ur Kisisel
Lecture notes by Ozg
Content: Modeling with first order equations. (section 2.3).
Suggested Problems:
2.3: 2, 3, 5, 13, 16, 19
1
Modeling with first order equations
Historically, interest in the subject of differen
MATH 219
Fall 2016
Lecture 4
ur Kisisel
Lecture notes by Ozg
Content: Differences between linear and nonlinear equations (section 2.4).
Suggested Problems:
2.4: 2, 6, 7, 10, 14, 22, 23
Recall that a first order ODE is called linear if it can be written i
Table of Annihilators and Integrals
1. The annihilator of xk is Dk+1 .
2. The annihilator of eax is D a.
3. The annihilator of xk eax is (D a)k+1 .
4. The annihilator of cos(bx) is D2 + b2 .
5. The annihilator of sin(bx) is D2 + b2 .
6. The annihilator of
DFERANSYEL DENKLEMLER 2008-2009 Gz Dnemi Diferansiyel Denklemlerin Snflandrlmas Birok mhendislik, fizik ve sosyal kkenli problemler matematik terimleri ile ifade edildii zaman bu problemler, bilinmeyen fonksiyonun bir veya daha yksek mertebeden trevlerini
Ch 2.7: Numerical Approximations: Eulers Method
Recall that a first order initial value problem has the form
dy = f (t , y ), y (t0 ) = y0 dt
If f and f / y are continuous, then this IVP has a unique solution y = (t) in some interval about t0. When the di
KNC DERECE DENKLEMLERN SER ZM
y = a n ( x x0 ) n
n =0
kuvvet serisi
m
1) Eer bir kuvvet serisinin ksmi toplamlar dizisinin lim m a n ( x x0 ) n limiti
n =0
var ise a n ( x x0 ) n kuvvet serisine x noktasnda yaknsak denir. Eer kuvvet serisi
n =0
bir x nok
LAPLACE DNM Lineer diferansiyel denklemlerin zmleri iin ok kullanlan yntemlerden birisi integral dnmleridir ve bir integral dnm
F ( s ) = K ( s, t ) f (t )dt
(1)
formundadr. Burada verilen bir f fonksiyonu F fonksiyonuna dnr ve F fonksiyonuna f in fonks
7. BRNC MERTEBEDEN LNEER DENKLEM SSTEMLER MATRSLER Birinci derece lineer diferansiyel denklem sistemlerinin zmnde matris kavram ve zellikleri doal olarak ortaya kmaktadr. Bir matris A(mxn) eklinde
a11 a A = 21 . a m1 . . a1n . . a2n . . . . . a mn
eklin
Ch 2.1: Linear Equations; Method of Integrating Factors
A linear first order ODE has the general form
dy = f (t , y ) dt
where f is linear in y. Examples include equations with constant coefficients, such as those in Chapter 1,
y = ay + b
or equations wit
Ch 2.2: Separable Equations
In this section we examine a subclass of linear and nonlinear first order equations. Consider the first order equation
dy = f ( x, y ) dx
We can rewrite this in the form
M ( x , y ) + N ( x, y ) dy =0 dx
For example, let M(x,y)
Ch 2.3: Modeling with First Order Equations
Mathematical models characterize physical systems, often using differential equations. Model Construction: Translating physical situation into mathematical terms. Clearly state physical principles believed to go
Ch 2.4: Differences Between Linear and Nonlinear Equations
Recall that a first order ODE has the form y' = f (t, y), and is linear if f is linear in y, and nonlinear if f is nonlinear in y. Examples: y' = t y - e t, y' = t y2. In this section, we will see
Ch 2.5: Autonomous Equations and Population Dynamics
In this section we examine equations of the form y' = f (y), called autonomous equations, where the independent variable t does not appear explicitly. The main purpose of this section is to learn how ge
Ch 2.6: Exact Equations & Integrating Factors
Consider a first order ODE of the form
M ( x, y ) + N ( x, y ) y = 0
Suppose there is a function such that
x ( x, y ) = M ( x, y ), y ( x, y ) = N ( x, y )
dy d + = [ x, ( x ) ] x y dx dx
and such that (x,y)