1. Second-order non-homogeneous linear
of (D2 + aD + b)y = r(x) in V . What we do is to
y(x) = c1 f1 (x) + c2 f2 + + ck fk
We turn our attention to the following nonhomogeneous case of 2nd-order linear ODE
for some unknown co
1. Higher order homogeneous linear
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
Proof. Clearly the quasi-polynomials xi ex are linP
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 y 00 + axy 0 + by = 0
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 condition
is forced to be z
1. Frobenius method
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 + a2 x2 + )
analytic at x = 0. Then the ODE
y2 (x) = y
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 = Ay, or just
points which can arise in a 2-dimensional
1. Separable ODEs
We start with the simplest form of ODE, called
the separable ODE.
By change of coordinates, (x, y) (x, u), the
picture transforms to
(x, u)-picture here
g(y)y 0 = f (x)
or we may equivalently write
g(y) dy = f (x) dx
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 probability density for some ranear ODEs is of course the
1. Ordinary Differential Equation (ODE)
2. Direction Field and Integral Curves
A first-order ODE is usually given by the followThe main subject of this course is Ordinary Difing form.
ferential Equation (called in short ODE).
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.
am (x x0 )m
= a0 + a1 (x x0 ) + a2 (x x
1. Examples and Applications
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
Bh0 = Av = 26.56A h
h0 = 26.56
1. Linear ODEs
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 look at the (algebraic) structure of the set of soluti
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 transformations L and L1 .
transform. The Laplace transform is a po