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.
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
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
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
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.