DIFFERENTIAL EQUATIONS FOR SCIENTISTS AND ENGINEERS
MATH 246H

Spring 2009
% Problem Set C % Problem 1a f = @(t,y)(exp(y)/(t.*exp(y)sin(y); % finding the approximate values [t, ya] = ode45(f, 0 : 1.5 : 3 , 1); format long; [t ya] % Graphing Solution on required interval o ode45(f , [0.5 4] , 1);
% 1b f = @(t,y)(exp(y)/(
DIFFERENTIAL EQUATIONS FOR SCIENTISTS AND ENGINEERS
MATH 246H

Spring 2009
% Problem 7a % to find the values at 1 1.5 and 3 I had to use ode45 twice because in %order for it to have the correct initial condition it needed to have the %interval start at 2, so I made two intervals; one starting at two % and going down to one
DIFFERENTIAL EQUATIONS FOR SCIENTISTS AND ENGINEERS
MATH 246H

Spring 2015
Howard Community College
Course Syllabus
Differential Equations
MATH 260
Course Description
Description: This course consists of concepts generally encountered in a first course in
differential equations. It introduces the basic techniques for solving fir
DIFFERENTIAL EQUATIONS FOR SCIENTISTS AND ENGINEERS
MATH 246H

Spring 2015
Ch 2.1: Linear Equations;
Method of Integrating Factors
A linear first order ODE has the general form
dy
= f (t , y )
dt
where f is linear in y. Examples include equations with
constant coefficients, such as those in Chapter 1,
y = ay + b
or equations wit
DIFFERENTIAL EQUATIONS FOR SCIENTISTS AND ENGINEERS
MATH 246H

Spring 2015
Ch 2.1: Linear Equations;
Method of Integrating Factors
A linear first order ODE has the general form
dy
f (t , y )
dt
where f is linear in y. Examples include equations with
constant coefficients, such as those in Chapter 1,
y ay b
or equations with var
DIFFERENTIAL EQUATIONS FOR SCIENTISTS AND ENGINEERS
MATH 246H

Spring 2015
Ch 1.1:
Basic Mathematical Models; Direction
Fields
Differential equations are equations containing derivatives.
The following are examples of physical phenomena involving
rates of change:
Motion of fluids
Motion of mechanical systems
Flow of current in e
DIFFERENTIAL EQUATIONS FOR SCIENTISTS AND ENGINEERS
MATH 246H

Spring 2015
[T,Y]=meshgrid(3:0.2:7,3:0.2:7);
S=Y.*(5Y);
L=sqrt(1+S.^2);
quiver(T,Y,1./L,S./L,0.5),axistight
title'DirectionFieldfordy/dt=Y*(5Y)'
Published with MATLAB?R2014b
0=y(5y) y=0,5
The equilibrium solutions of dy/dt=y(5y) are 0 and 5. As it is shown in the
DIFFERENTIAL EQUATIONS FOR SCIENTISTS AND ENGINEERS
MATH 246H

Spring 2015
August 2, 2012 13:11
bansw
Sheet number 1 Page number 739
cyan black
Answers to
Problems
C H A P T E R
1
Section 1.1, page 7
1.
3.
5.
7.
9.
11.
12.
13.
14.
15.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.
y 3/2 as t
2. y diverges from 3/2 as t
y
DIFFERENTIAL EQUATIONS FOR SCIENTISTS AND ENGINEERS
MATH 246H

Fall 2014
Tuesday,
Nov 26, 2013
Midterm 3
Math 246H
1. A mass of m=2grams is hung vertically from a spring. At rest it stretches the spring
x=3cm. At time t = 0, the mass is pulled h0 =3cm away from its resting position.
Suppose the spring moves in a medium where t
DIFFERENTIAL EQUATIONS FOR SCIENTISTS AND ENGINEERS
MATH 246H

Fall 2014
Thursday,
Nov 20, 2014
Midterm 3
Math 246H
1. A mass of m=0.25 slugs stretches a spring x=3 inches. Suppose the mass moves in a medium
which exerts a dampening force of f =8 pound when the mass moves at a speed of s=2 feet
per second. Recall that in these
DIFFERENTIAL EQUATIONS FOR SCIENTISTS AND ENGINEERS
MATH 246H

Fall 2014
Thursday,
Sept 26, 2013
Midterm 1
Math 246H
1. Consider the dierential equation
dy
= 2yt.
dt
5 P.
(a) Solve this equation with the initial condition y(0) = 1.
Solution: The equation is separable, so all solutions are either stationary or obtained by
separ
DIFFERENTIAL EQUATIONS FOR SCIENTISTS AND ENGINEERS
MATH 246H

Fall 2014
Thursday,
Oct 23, 2014
10 P.
Midterm 2
Math 246H
1. Use the Euler method with step size h = 1 to approximate y(3), where y(t) is the unique
solution to the initial value problem
y = ety ,
y(1) = 0.
Solution: Using the approximation y(t + h) y(t) hy (t) =
DIFFERENTIAL EQUATIONS FOR SCIENTISTS AND ENGINEERS
MATH 246H

Fall 2014
Thursday,
Sept 25, 2014
Midterm 1
Math 246H
1. Find a general solution to the dierential equations below. Solve the stated initial value problem, and give the interval of denition for the solution.
10 P.
(a)
dy
2
= y + 1,
dt
t
y(1) = 2
Solution: Using the
DIFFERENTIAL EQUATIONS FOR SCIENTISTS AND ENGINEERS
MATH 246H

Spring 2015
Ch 1.1:
Basic Mathematical Models; Direction Fields
Differential equations are equations containing derivatives.
The following are examples of physical phenomena
involving rates of change:
Motion of fluids
Motion of mechanical systems
Flow of current in e
DIFFERENTIAL EQUATIONS FOR SCIENTISTS AND ENGINEERS
MATH 246H

Spring 2015
Ch 2.4: Differences Between Linear and
Nonlinear Equations
Recall that a first order ODE has the form y' = f (t, y), and is
linear if f is linear in y, and nonlinear if f is nonlinear in y.
Examples: y' = t y  e t, y' = t y2.
In this section, we will see
DIFFERENTIAL EQUATIONS FOR SCIENTISTS AND ENGINEERS
MATH 246H

Spring 2015
Ch 6.1: Definition of Laplace Transform
Many practical engineering problems involve mechanical or
electrical systems acted upon by discontinuous or impulsive
forcing terms.
For such problems the methods described in Chapter 3 are
difficult to apply.
In th
DIFFERENTIAL EQUATIONS FOR SCIENTISTS AND ENGINEERS
MATH 246H

Spring 2009
Joey Hartstein, Kelley Heffner Problem Set E a) This is the code for solving and graphing the differential equation, dsolve is used to solve the equation and ezplot is used to plot it
ode12a = 'D2y + 2*Dy + 2*y = sin(t)'; sol12a = dsolve(ode12a, 'y(0
DIFFERENTIAL EQUATIONS FOR SCIENTISTS AND ENGINEERS
MATH 246H

Spring 2009
% % Problem Set E %Do problems 12,13abc,17acd,18 from the Problem Set E in DEwM pp193205 %Note: the equation in 12b) shoud read y'+2y'+2y=0 and in 13a) h(t) should % %be 1 on pi<=t<10 and 0 otherwise. ode12a = 'D2y + 2*Dy + 2*y = sin(t)'; sol12a = d
DIFFERENTIAL EQUATIONS FOR SCIENTISTS AND ENGINEERS
MATH 246H

Spring 2015
Ch 3.6: Nonhomogeneous Equations;
Method of Undetermined Coefficients
Recall the nonhomogeneous equation
y p (t ) yq (t ) y g (t )
where p, q, g are continuous functions on an open interval I.
The associated homogeneous equation is
y p (t ) yq(t ) y 0
In
DIFFERENTIAL EQUATIONS FOR SCIENTISTS AND ENGINEERS
MATH 246H

Spring 2015
Ch 3.5: Nonhomogeneous Equations;
Method of Undetermined Coefficients
Recall the nonhomogeneous equation
y + p (t ) y + q (t ) y = g (t )
where p, q, g are continuous functions on an open interval I.
The associated homogeneous equation is
y + p (t ) y + q
DIFFERENTIAL EQUATIONS FOR SCIENTISTS AND ENGINEERS
MATH 246H

Spring 2015
Ch 3.7: Mechanical & Electrical Vibrations
Two important areas of application for second order linear
equations with constant coefficients are in modeling
mechanical and electrical oscillations.
We will study the motion of a mass on a spring in detail.
An
DIFFERENTIAL EQUATIONS FOR SCIENTISTS AND ENGINEERS
MATH 246H

Spring 2015
Ch 7.5: Homogeneous Linear Systems with
Constant Coefficients
We consider here a homogeneous system of n first order linear
equations with constant, real coefficients:
x1
a11 x1 a12 x2 a1n xn
x2
a21 x1 a22 x2 a2 n xn
xn
an1 x1 an 2 x2 ann xn
This system c
DIFFERENTIAL EQUATIONS FOR SCIENTISTS AND ENGINEERS
MATH 246H

Spring 2015
Ch 6.1: Definition of Laplace Transform
Many practical engineering problems involve mechanical or
electrical systems acted upon by discontinuous or impulsive
forcing terms.
For such problems the methods described in Chapter 3 are
difficult to apply.
In th
DIFFERENTIAL EQUATIONS FOR SCIENTISTS AND ENGINEERS
MATH 246H

Spring 2015
Ch 3.1: Second Order Linear
Homogeneous Equations with Constant
Coefficients
A second order ordinary differential equation has the
general form
y = f (t , y , y )
where f is some given function.
This equation is said to be linear if f is linear in y and y
DIFFERENTIAL EQUATIONS FOR SCIENTISTS AND ENGINEERS
MATH 246H

Spring 2015
Ch 3.1: Second Order Linear Homogeneous
Equations with Constant Coefficients
A second order ordinary differential equation has the
general form
y f (t , y , y
)
where f is some given function.
This equation is said to be linear if f is linear in y and y'
DIFFERENTIAL EQUATIONS FOR SCIENTISTS AND ENGINEERS
MATH 246H

Spring 2015
Ch 3.7: Mechanical & Electrical Vibrations
Two important areas of application for second order linear
equations with constant coefficients are in modeling
mechanical and electrical oscillations.
We will study the motion of a mass on a spring in detail.
An
DIFFERENTIAL EQUATIONS FOR SCIENTISTS AND ENGINEERS
MATH 246H

Fall 2014
Thursday,
Sept 26, 2013
1. Let D =
5 P.
d
dt .
Midterm 1
Math 246H
Find a general solution to the dierential equations below:
(a) D2 y 4Dy + 5 = 0
Solution: The characteristic polynomial is x2 4x + 5. By the quadratic formula, the
roots are
4 42 4 5
= 2 i