Finding Particular Solutions to Boundary-value Problems Using Fourier Methods
Final Exam Study Guide
December 7, 2009
1 Introduction
Objective 1.
Be able to derive the general solution of the differential equation
y
=
ay
+
b
where
a
and
b
are constants. Do not forget the equilibrium solution!
Problem 1.1.
A falling object satisfies the initial value problem
d
v
d
t
= 9
.
8
−
(
v/
5)
,v
(0) = 0
where
v
denotes velocity. Find the time that must elapse for the object to reach 98% of its limiting velocity. How far does the object fall in that time?
Objective 2.
Be able to classify differential equations with respect to
•
Ordinary versus Partial
•
Linear versus Nonlinear
•
Order
Problem 2.1.
Classify the following equations. 1.
t
2 d
2
y
d
t
2
+
t
d
y
d
t
+ 2
y
= sin
t
2.
d
2
y
d
t
2
+ sin(
t
+
y
) = sin
t
3.
d
3
y
d
t
3
+
t
d
y
d
t
+ (cos
2
t
)
y
=
t
3
1
2 First Order Differential Equations
Objective 3.
Solve first order linear equations by the method of integrating factors.
Problem 3.1.
Find the general solutions to the following equations. 1.
y
−
y
= 2
te
2
t
2.
y
−
2
y
= 3
e
t
3.
y
+
y
= 5sin2
t
Problem 3.2.
Find the value for
y
0
for which the solution to the initial value problem
y
+ 2 3
y
= 1
−
1 2
t, y
(0) =
y
0
touches but does not cross the
t
-axis.
Objective 4.
Recognize and solve separable differential equations.
Problem 4.1.
Find the solutions of the following separable equations. 1.
d
y
d
x
=
x
−
e
−
x
y
+
e
y
2.
xy
= (1
−
y
2
)
1
/
2
3.
sin2
x
d
x
+ cos3
y
d
y
= 0
Problem 4.2.
Solve the initial value problem
y
= (1 + 3
x
2
)
/
(3
y
2
−
6
y
)
, y
(0) = 1
and determine the interval in which the solution is valid.
Objective 5.
Recognize and solve exact equations, and be able to use integrating factors to make an equation exact.
Problem 5.1.
Solve the following differential equations, if they are exact. 1.
2
x
+
y
2
+ 2
xyy
= 0 2
2.
(2
x
+ 4
y
) + (2
x
−
2
y
)
y
= 0
3.
(2
x
+ 3) + (2
y
−
2)
y
= 0
4.
x
d
x
(
x
2
+
y
2
)
3
/
2
+
y
d
y
(
x
2
+
y
2
)
3
/
2
= 0
.
Problem 5.2.
Find an integrating factor for the equation
(3
xy
+
y
2
) + (
x
2
+
xy
)
y
= 0
and solve it.
Objective 6.
Understand the existence and uniqueness of solutions to first order equations.
3 Second Order Linear Equations
Objective 7.
Solve second order homogeneous equations with constant coefficients by interpreting the characteristic polynomial.
Problem 7.1.
Find the general solution of the following differential equations. 1.
y
+ 5
y
+ 6
y
= 0
2.
4
y
−
y
= 0
3.
y
+ 5
y
= 0
4.
y
2
y
+ 2
y
= 0
5.
y
+ 6
y
+ 9 = 0
Objective 8.
Understand the existence and uniqueness of solutions to second order linear initial value problems and the principle of superposition.
Problem 8.1.
Show that any linear combination of
sin(
x
)
and
cos(
x
)
can be written as
α
sin(
x
+
β
)
for some
α
and
β
.
3
Finding Particular Solutions to Boundary-value Problems Using Fourier Methods
Source: https://id.scribd.com/document/77768948/Differential-Equations-Review