The Third Exam Review
1. Use the denition of the Laplace transform to show that L[t] =
2. Find the Laplace transform of the following functions.
a) t4 e2t + cos 5t 7
3 et d
sin(2 ) cos(2t 2 )d
3. Find the inverse Lap
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
y = 2 + ex is its solution.
(b) Classify the equation y 3y + 2y = 0
First Order Dierential Equations
Introduction to Dierential Equations; Classications of Dierential
A dierential equation is an equation in unknown function that contains one or more derivatives of the unknown function.
Matlab Notes for Differential Equations (MA320)
Basics of first order differential equations
Numerical solutions using ode45
Practice problems 1
6. Second and higher order differential eq
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
Nonhomogeneous Linear Dierential Equations. Methods.
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
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
Systems of Dierential Equations
A system of rst order dierential equations in two unknown functions x and y has the form
= f (x, y, t),
= g(x, y, t).
It may be helpful to think that the independent variable t denotes the time
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
Review of Systems of ODEs
1. Find all the equilibrium points of the following systems.
= 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