Math 320
Lia Vas
The Third Exam Review
1. Use the denition of the Laplace transform to show that L[t] =
1
.
s2
2. Find the Laplace transform of the following functions.
a) t4 e2t + cos 5t 7
b)
t
0
3 et d
c)
t
0
sin(2 ) cos(2t 2 )d
3. Find the inverse Lap
Math 320
Lia Vas
First Exam Review
1. Checking if a given function is a solution. Classication.
(a) Classify the equation y + 3x2 y = 6x2 based on the order and linearity and show that
3
y = 2 + ex is its solution.
(b) Classify the equation y 3y + 2y = 0
Math 320
Lia Vas
First Order Dierential Equations
Introduction to Dierential Equations; Classications of Dierential
Equation
A dierential equation is an equation in unknown function that contains one or more derivatives of the unknown function.
The order
Matlab Notes for Differential Equations (MA320)
Lia Vas
Content
1.
2.
3.
4.
5.
M-files
Basics of first order differential equations
Direction field
Numerical solutions using ode45
Euler method
Practice problems 1
6. Second and higher order differential eq
Math 320
Lia Vas
Second and Higher Order Linear Dierential Equations
A second order dierential equation is linear if it can be written in the form
a(x)y + b(x)y + c(x)y = g(x).
The general solution of such equation will depend on two constants. An initial
Math 320
Lia Vas
Nonhomogeneous Linear Dierential Equations. Methods.
Applications
Consider a nonhomogeneous linear equation
an y (n) + an1 y (n1) + . . . + a0 y = g(x).
The general solution of such equation is of the form
y = yh + yp
where yh is the gene
Math 320
Lia Vas
The Laplace Transform
The Laplace transform is an integral operator (meaning that it is dened via an integral and that
it maps one function to the other). It can be useful when solving dierential equations because it
transforms a dierenti
Math 320
Lia Vas
Systems of Dierential Equations
A system of rst order dierential equations in two unknown functions x and y has the form
dx
= f (x, y, t),
dt
dy
= g(x, y, t).
dt
It may be helpful to think that the independent variable t denotes the time
Math 320
Lia Vas
The Second Exam Review
1. Homogeneous equations with constant coecients. Solve the following equations.
(a) y 2y + 5y = 0
(b) y 2y + y = 0
(c) y (4) y = 0
(d) y (4) 5y 36y = 0
(e) y (5) 32y = 0.
(f) y (5) + 32y = 0.
2. Non-homogeneous equ
Math 320
Lia Vas
Review of Systems of ODEs
1. Find all the equilibrium points of the following systems.
(a)
dx
dt
(b)
dx
dt
dy
dt
dy
dt
= x x2 xy
= x x2 xy
= 0.75y y 2 0.5xy
= 0.5y 0.25y 2 0.75xy
2. The point (0,0) is the only equilibrium point of the fol