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
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
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)
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 2.4: 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 2.3: 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 2.2: 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 2.1: 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
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
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
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
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