Math 208 - Exam 2, March 17, 2009
Name:
Key
Problem 1. Find the general solution on R to each of the following dierential equations:
(a). y (4) + 2y + y = 0.
(b). y (4) 8y + 16y = sin t + e2t .
(c). y (4) + 2y = t
(d). y (3) + y y = te2t
Solution 1. (a).
Jenny Gilbert
MATH 208-01
Final Review Outline: Major Goals in Differential Equations
Goal: Be able to define and understand differential equations, associated terminology,
and real-life applications
o Differential equation
Composed of 3 key elements
An
Jenny Gilbert
MATH 208-01
Midterm Review
Outline: Major Goals in Differential Equations
Be able to define and understand differential equations, associated terminology, and reallife applications
o Differential equation
Composed of 3 key elements
An equ
Math 208 - Quiz 10, April 9, 2009
Name:
Key
Problem 1. Use the Laplace transform to solve the initial value problem: y 4y + 4y = 0, y (0) = 1, y (0) = 1.
Solution 1. By using the formula Lcfw_f (n) (t) = sn Lcfw_f (t) n=1 snk f (k1) (0) we may take the La
Math 208 - Quiz 9, April 2, 2009
Name:
Key
Problem 1. Find the general solution to the dierential equation x2 y 3xy + 4y = 0, for x > 0.
Solution 1. We rst look for a solution of the form y (x) = xr for some r R. In this case y = rxr1 and
y = r(r 1)xr2 he
Math 208 - Quiz 8, March 26, 2009
Name:
Key
Problem 1. Find a recurrence relation for the coecients in a power series solution to dierential equation y + y
xy = 0 about the point x0 = 0. Use this to calculate the rst 4 coecients to a solution which satis
Math 208 - Quiz 7, March 12, 2009
Name:
Key
Problem 1. Find the general solution on R to the dierential equation y (6) y = 0.
Solution 1. The characteristic polynomial to this equation is r6 1. To nd the 6 roots of this polynomial we use
k
Eulers identity
Math 208 - Quiz 6, February 26, 2009
Name:
Key
Problem 1. Find the general solution on R to the dierential equation y 2y + y = et .
Solution 1. First of all note that the characteristic polynomial of the associated homogeneous equation y 2y + y = 0
is r2
Math 208 - Quiz 5, February 19, 2009
Name:
Key
Problem 1. Find a solution on R to the dierential equation y + 2y + 5y = 0 subject to the initial conditions
y (0) = 1, y (0) = 0.
Solution 1. The characteristic polynomial associated to this equation is r2 +
Math 208 - Quiz 4, February 5, 2009
Name:
Key
Problem 1. Find the solution to the dierential equation (2y sin x ex sin y ) + (ex cos y 2 cos x)y = 0.
Solution 1. This dierential equation is of the form M (x, y ) + N (x, y )y = 0 where My = 2 sin x ex cos
Math 208 - Quiz 3, January 29, 2009
Name:
Key
Problem 1. A large tank is half full with a mixture of salt dissolved in water. Pure water is pumped into the tank
at the rate of 3 gallons per minute, and water is let out of the tank at the rate of 1 gallon
Math 208 - Quiz 1, January 15, 2009
Name:
Key
Problem 1. Solve each of the following initial value problems and graph the solution.
(a).
dy
dt
= y + 3, y (0) = 2.
(b).
dy
dt
= 2y + 1, y (0) = 1.
(c).
dy
dt
= y 2, y (0) = 2.
Solution 1. Ill show how to sol
Math 208 - Exam 3, April 14, 2009
Name:
Key
Problem 1 (20 points). Find the general solution to the dierential equation y + xy = 1 + x in powers of
x. You may leave your answer in the form of a recurrence relation for the coecients of y , in any case
howe
Section 9.4: Systems of Linear Differential Equations
Normal Form, Fundamental Solution Set, Fundamental Matrix
o Normal Form
Homogenous when , otherwise it is nonhomogeneous
o Fundamental Solution Set
A set of solutions that are linearly independent o