Math 224
Final Review Sheet
The exam is comprehensive and will cover all the material which we discussed from chapters 1 – 5, and B1, B2, B3.
The following is a list of the topics which you may expect to appear on the final exam:
Differential Equations:
1.
First Order Differential Equations
a.
b.
c.
d.
Separable Differential Equations and their solution.
Homogeneous Differential Equations and their solution.
Linear Differential Equations and their solution.
Euler’s Method
2.
Homogeneous equations with constant coefficients.
3.
Nonhomogeneous equations: The method of Annihilators.
4.
Systems of differential equations, Using;
a. Eigenvalues and Eigenvectors method.
b. D-Operator.
5.
Application-Problems:
a.
b.
c.
d.
e.
Circuit problems
Mixture problems
Free fall problems
Simple harmonic problems
Newton’s law of cooling problems
Linear Algebra:
1.
2.
3.
4.
5.
6.
7.
Basic operations with matrices
Determinants
Linear combinations and linear independence
Spanning sets
Finding Bases for the row-space, Null-space and the column space of a matrix.
Solving system of linear equations using matrices
Row Echelon and Reduced Row Echelon form of a matrix and the rank of a matrix.
Sample Problems:
1. Solve the following first order differential equations:
dy
− y 2 = 1 Subject to y(0)=1
dx
a.
( x 2 + 1)
b.
y ′ = (sin x sec x ) y − 2 sin x
c.
( x 2 − 1) y′ + 2 x y 2 = 0 subject to y ( 2) = 1
d.
x y′ + (1 + x ) y = e− x sin(2 x)
2. Solve the following homogeneous differential equations: Note that y = y ( x )
3.
4.
5.
6.
a.
( D 2 + 8 D + 12) y = 0
b.
y′′ − 8 y′ + 16 y = 0
c.
y′′ − 4 y ′ + 13 y = 0
d.
y (5) − 2 y (4) + y (3) = 0
Use the method of the annihilator to solve the following non-homogeneous differential equations: Hint :
a.
( D 2 + 1) y = 3cos x + 4sin x
b.
y′′ + 4 y′ + 4 y = 5 x e−2 x
yG = yC + yP
System of differential equations:
a.
 x1′ (t ) = 5 x1 (t ) − x2 (t )

 x2′ (t ) = 3x2 (t )
D-Operator method
b.
 x1′ (t ) = 6 x1 (t ) − x2 (t )

 x2′ (t ) = −5 x1 (t ) + 2 x2 (t )
Using the eigenvalues/eigenvector method
Matrices:
 x1 + 2 x2 + x3 = 1

3x1 + 5 x2 + x3 = 3
2 x + 6 x + 7 x = 1
2
3
 1
a.
Using Gaussian-elimination method solve
b.
Find the bases for the column space and the row space of the matrix
c.
 3  −1 0 
 1  1 
     
2
3
Determine if the given set of vectors {  ,  } in ℝ , and { 0 , 0 , 1 } in ℝ form a basis.






 −1  2 
0   1  1 
d.
Find a basis for the row space, column space, and the null space of the matrix
 1 1 −3 2 


 3 4 −11 7 
1 3 1 
A=

 2 1 −1
Euler’s Method:
By hand, carry out three steps of Euler's method for the differential equation
with initial condition y(0) = 1/2 and a step size of 0.1
dy
=y−x
dx
[Show work]
7.
Application problems:
a.
In the investigation of a homicide, the time of death is important. The normal body
temperature of most healthy people is 98.6
0
0
F . Suppose that when a body is discovered at noon,
its temperature is 82 F . Two hours later it is 72
what was the approximate time of death?
0
F
. If the temperature of the surroundings is
650 F
,
b.
Simple harmonic motion problems with or without the damping force.
c.
A Five-foot-tall boy tosses a tennis ball straight up from the level of the top of his head. Neglecting frictional forces, if the object
hits the ground 8 seconds after the boy releases it, find the time when the ball reaches its maximum height and the maximum
2
height of the ball. [Hint: Solve
d y
dt
2
d.
Circuit Problem.
e.
Simple harmonic motion problems.
= − g where g = 32 ft / sec 2 and y is the height of the ball at any given time t.]