6
Exact solutions to second-order line dierential equations with boundary conditions
Example. We study the boundary value problem
u + 2u + 5u = 0;
u(0) = 2, u(/4) = 1.
First, we look for a special solution of form u(x) = erx . The characteristic equation
1
Power series solutions to second-order linear dierential equations
at ordinary points
Example 1: y = y.
Set y(x) = an xn , then we have y (x) = an nxn1 and thus,
n=0
n=0
n1
an nx
=
n=0
Note that (replacing n by n 1)
n
an x =
Comparing the coecients of
a
3
Fourier series solutions to the heat equations with initial-boundary
conditions
Let us start with a simple scalar linear dierential equation
du
= au.
dt
Its solution is given by u(t) = u(0)eat , where u(0) is the initial value at t = 0.
Next, we study t
7
Stability analysis of critical numbers of nonlinear dierential systems
We start with a simple exponential growth/decay model:
N (t) = bN (t) N (t),
where N (t) denotes the population size/density at time t, and r represents the birth rate, is the death
5
Fourier series solutions to the Laplaces equations with boundary
conditions
We study the Laplaces equation for a rectangle:
u =
2u 2u
+ 2 = 0, 0 < x < L1 , 0 < y < L2
x2
y
together with the boundary conditions
u
(0, y) = 0, 0 y L2 ;
x
u
(L1 , y) = 0, 0
2
Power series solutions to second-order linear dierential equations
at regular singular points
Let us study the equation: 2x2 y + 3xy + (x 1)y = 0. Assume y(x) =
2an n(n 1)xn +
n=0
3an nxn +
n=0
an xn+1
n=0
n
n=0 an x ,
we obtain
an xn = 0.
n=0
Simplify
4
Fourier series solutions to the wave equations with initial-boundary
conditions
Example: u (t) = u(t).
Case 1: if > 0, we have u(t) = ae
t
+ be
t ;
Case 2: if = 0, we have u(t) = a + bt;
Case 3: if < 0, we have u(t) = a cos( t) + b sin( t).
To determine
Name:
Student Id:
MA550-70 Dierential Equations II Quiz 7
1. Find all the critical/equilibrium points for the following system and discuss the stability of the corresponding equilibrium solutions.
dx1
= x1 + x2 3;
dt
dx2
= x2 + x3 5;
dt
dx3
= x3 + x1 4.
d
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Student Id:
MA550-70 Dierential Equations II Quiz 6
1. Find the exact solution to the boundary value problem
u (x) + u(x) = 3 cos(2x);
u(0) = 0, u() + u () = 0.
Solution. From the characteristic equation r2 + 1 = 0, we obtain two linearly independen
Name:
Student Id:
MA550-70 Dierential Equations II Quiz 2
1
2
2
Find the power series solution about x = 0 to the dierential equation x y + xy + (x )y = 0.
4
Solution. Note that in the standard form y + p(x)y + q(x)y = 0, we have p(x) = 1/x and q(x) =
1
Name:
Student Id:
MA550-70 Dierential Equations II Quiz 3
1. Find the Fourier series solution to the heat equation
u
2u
= 8 2 , t > 0, 0 < x < 2
t
x
with Dirichlet boundary condition
u(t, 0) = u(t, 2) = 0, t > 0
and initial condition
u(0, x) = 2x x2 , 0 <
Name:
Student Id:
MA550-70 Dierential Equations II Quiz 1
Find the power series solution about x = 0 to the dierential equation (1 x2 )y 2xy + 2y = 0.
Solution. Let y(x) = an xn , we have y (x) = an nxn1 and y (x) = an n(n 1)xn2 .
n=0
n=0
n=0
Substituting