ln06-ode-19mar07
1. Second-order non-homogeneous linear
ODEs
1
of (D2 + aD + b)y = r(x) in V . What we do is to
write
y(x) = c1 f1 (x) + c2 f2 + + ck fk
We turn our attention to the following nonhomog
ln07-ode-26mar07
1. Higher order homogeneous linear
ODEs
An n-th order homogeneous linear ODE can be
given in the following standard form.
(1) y (n) +pn1 (x)y (n1) + +p1 (x)y 0 +p0 (x)y = 0
1
Proof. C
ln05-ode-14mar07
1
1. Euler-Cauchy equations
this not a coincidence because we can transform the
Euler-Cauchy equation to the one with constant coAn ODE of the form
efficients. For x = et , we have
x2
ln11-ode-30apr07
1
the boundary condition forces c1 = c2 = 0. For
= 0 the general solution is the linear polynomial
The following form of equation is called a y = c x+c which again by the boundary co
ln10-ode-23apr07
1. Frobenius method
1
If the indicial equation has a double root r1 = r2 =
r, then the solutions are of the form
Theorem 1 (Frobenius Method). Let b(x), c(x) be
y1 (x) = xr (a0 + a1 x
ln08-ode-09apr07
1
In this section we assume tr A = a11 + a22 0,
since the other cases with positive trace are covIn this section we work on the types of critical ered by considering the equation y0 =
ln02-ode-28feb07
1. Separable ODEs
We start with the simplest form of ODE, called
the separable ODE.
1
By change of coordinates, (x, y) (x, u), the
picture transforms to
(x, u)-picture here
g(y)y 0 =
ln13-ode-09may07
1
1. Derivatives
Besides for Laplace transforms, there also is an
application of convolution in probability theory.
What makes the Laplace transform work for lin- Let f (t) be the pro
ln01-ode-21feb07
1. Ordinary Differential Equation (ODE)
1
2. Direction Field and Integral Curves
A first-order ODE is usually given by the followThe main subject of this course is Ordinary Difing for
ln09-ode-16apr07
1. Power series method
where p(x), q(x) are analytic functions. Then
A power series is a method to represent an analytic function by a converging sequence of polynomials as follows.
y
ln04-ode-12mar07
1. Examples and Applications
we have
So far we discussed various methods to solve firstorder ODEs. Before going on for the second order
ODEs, we give some examples of applying them to
ln03-ode-07mar07
1
1. Linear ODEs
R
e
An equation of the form
y 0 + p(x)y = r(x)
is called a linear ODE of first order.
We want a general solution to this ODE. For this
purpose, we are going to take a
ln12-ode-07may07
1
We can think that in this method of Laplace transform for ODEs, the process of integration is nicely
In the rest of this course, we will work on Laplace hidden into the transformati