Sample Midterm
Math 046 (020) Introduction to Ordinary Dierential Equations
Mathematics Department
University of California, Riverside
Fall Quarter 2012
Name:
1. Solve the initial-value problem
y 0 + 5y = ex ; y(0) = 2
2. Solve the dierential equation
y0
Chapter5(continue):ExactEquations,
IntegratingFactors
Consider a first order differential equation of the form
M ( x, y ) N ( x, y ) y0
Suppose there is a function F such that
Fx ( x, y ) M ( x, y ), Fy ( x, y ) N ( x, y )
and such that F(x,y) = c define
Chapter5:ExactFirstOrderDifferential
Equations
Many times, it will be convenient to write a first
order ordinary differential equation in the form
M ( x, y )dx N ( x, y )dy 0.
It is the same as M(x,y)+N(x,y)y=0 or
N ( x, y ) y ' M ( x, y ).
The reason is
Chapter11:TheMethodofUndetermined
Coefficients
Let
L( y ) y p( x) y q( x) y,
then the general solution of the equation
L( y ) f ( x) is yh y p ,
where is the general solution of
yha particular solution of
L( y ) 0
and
is
yp
L( by
y)
f ( x).9. We just nee
Chapter 9: Second-Order Linear Homogeneous
Differential Equations with Constant Coefficients
Thus far we have concentrated on first-order
differential equations. We will now turn our
attentions to the second-order case.
We call a second-order equation
y f
Characteristic Equation
In general, to solve the second order equation with constant
coefficients,
ay by cy 0,
we begin by assuming a solution of the form y = ert.
Substituting this into the differential equation, we obtain
ar 2 e rt bre rt ce rt 0
Simpli
Chapter4:SeparableEquations
A first order equation y=f(x, y) is separable if
f ( x, y ) g ( x) / h( y ).
That is,
h( y )dy g ( x)dx
By integration, we get:
h( y )dy g ( x)dx.
Chapter4:SeparableEquations
Another way to write the separable equation is:
A( x
Chapter1:
BasicConcepts
A differential equation is an equation
containing one or more derivatives of an
unknown function.
Example 1:
y ' 5 x 3
e y y"2( y ' ) 2 1
(1.1)
(1.2)
Chapter1:
BasicConcepts
A differential equation is an ordinary
differential equat
Chapter6(continue):BernoulliEquations
Consider a first order differential equation of the form
y ' p( x) y q( x) y r r 1.
We have:
y'
1 r
p ( x) y q( x) or y 0 if r 0.
r
y
Let
z y1 r
then
z '(1 r ) p( x) z (1 r )q( x).
is a first order linear equation wi
Chapter7:Application
GrowthandDecay
One of the most common mathematical
models for a physical process is exponential
model Q=aQ. The rate of change is
proportional to the quantity.
a (t t )
Q
Q
e
.
Solution:
0
k (t t )
Q
'
kQ
,
Q
Q
e
.
Radioactive Decay:
Sample Final
Math 046 (020) Introduction to Ordinary Dierential Equations
Mathematics Department
University of California, Riverside
Fall Quarter 2012
Name:
1. Solve the dierential equation
y 000
2y 00
y 0 + 2y = 0
2. Solve the initial-value problem
y 00
Homework 7
Math 046 (020) Introduction to Ordinary Dierential Equations
Due Thursday, December 6 in Discussion
Mathematics Department
University of California, Riverside
Fall Quarter 2012
The Method of Undetermined Coefficients
11.23. Find the general sol
Homework 4
Math 046 (020) Introduction to Ordinary Dierential Equations
Due Thursday, November 1 in Discussion
Mathematics Department
University of California, Riverside
Fall Quarter 2012
Linear First-Order Dierential Equations
6.35. Solve the dierential
Math 46 Practice Final Exam
Dec 7, 2016
NAME (please print):
NetID (please print):
1. Write clearly and show all work.
2. You will be graded for both completeness and correctness for this exam.
3. This is a closed book test and no calculators are allowed.
Math 46 Practice Final Exam Solution
Dec 7, 2016
NAME (please print):
NetID (please print):
1. Write clearly and show all work.
2. You will be graded for both completeness and correctness for this exam.
3. This is a closed book test and no calculators are
Math 46 Practice Final Exam
Dec 7, 2016
NAME (please print):
NetID (please print):
1. Write clearly and show all work.
2. You will be graded for both completeness and correctness for this exam.
3. This is a closed book test and no calculators are allowed.
Math 46 Practice Final Exam Solution
Dec 7, 2016
NAME (please print):
NetID (please print):
1. Write clearly and show all work.
2. You will be graded for both completeness and correctness for this exam.
3. This is a closed book test and no calculators are
Homework 1
Math 046 (020) Introduction to Ordinary Dierential Equations
Due Thursday, October 11 in Discussion
Mathematics Department
University of California, Riverside
Fall Quarter 2012
Instructions: Write up your solutions neatly on separate sheets of
Homework 2
Math 046 (020) Introduction to Ordinary Dierential Equations
Due Thursday, October 18 in Discussion
Mathematics Department
University of California, Riverside
Fall Quarter 2012
Instructions: Write up your solutions neatly on separate sheets of
Homework 5
Math 046 (020) Introduction to Ordinary Dierential Equations
Due Thursday, November 15 in Discussion
Mathematics Department
University of California, Riverside
Fall Quarter 2012
Linear Dierential Equations: Theory of Solutions
8.40. Find the Wr
Homework 3
Math 046 (020) Introduction to Ordinary Dierential Equations
Due Thursday, October 25 in Discussion
Mathematics Department
University of California, Riverside
Fall Quarter 2012
Instructions: Write up your solutions neatly on separate sheets of
Homework 6
Math 046 (020) Introduction to Ordinary Dierential Equations
Due Thursday, November 29 in Discussion
Mathematics Department
University of California, Riverside
Fall Quarter 2012
nth-Order Linear Homogeneous Dierential Equations with Constant Co
Chapter6:LinearFirstOrderEquations
A first order equation y=f(x, y) is linear if f(x,y)=
-p(x)y+q(x),
i.e., y=-p(x)y+q(x) or y+p(x)y=q(x).
Or
( p ( x) y q ( x)dx dy 0.
We have:
M (linear
x, y )if
p ( x ) y q ( x), N ( x, y ) 1.
It is homogeneous
q(x)=0.