Worksheet 2 Solutions
Solve the following equations:
1. y + 4y + 3y = 0 with y(0) = 2 and y (0) = 1.
The characteristic equation is r 2 +4r+3 = 0, and the two roots are r =
1 and r = 3. Therefore the general solution is y = c1 ex + c2 e3x .
Initial condit
Worksheet 3: Method of undetermined coecients
Consider the motion of a harmonic oscillator governed by the equation
x + 2x + 2x = F (t)
where x(t) is the displacement from the rest position of oscillator. F (t)
represents the external driving force applie
Worksheet 3 Solutions
Consider the motion of a harmonic oscillator governed by the equation
x + 2x + 2x = F (t)
where x(t) is the displacement from the rest position of oscillator. F (t)
represents the external driving force applied to the oscillator.
1.
Worksheet 2: Homogeneous, linear, second order
equations with constant coecients
Solve the following equations:
1. y + 4y + 3y = 0 with y(0) = 2 and y (0) = 1.
2. y + 2y + 8y = 0 with y(0) = 1 and y (0) = 0.
3. 9y 12y + 4y = 0 with y(0) = 2 and y (0) = 1.
Worksheet 1: First order equations
For the following equations, determine whether they are linear or nonlinear
and then solve each of them.
1. xy + 2y = sin x.
2. y = x2 /y.
3. 2xy 2 + 2y + 2x2 y + 2x y = 0.
1
Worksheet 4 solutions
Solve y + 2 y = cos 2t where 2 = 4 with Laplace transform. The initial
conditions are y(0) = 1 and y (0) = 0.
Transforming the equation and applying the initial conditions gives
s
s+4
s
2
2
+s
(s + s)Y = 2
s +4
1
s
s
2 5
Y = 2
+ 2
2
Worksheet 5: Delta function
Consider the harmonic oscillator with an impulsive force y + y = (t 1).
The initial conditions are y(0) = 0 and y (0) = 0.
Solve this problem using the Laplace transform.
The homogeneous solution of the problem is yh = c1 cos
Worksheet 6: Eulers equation
1. Solve x2 y + 4xy + 2y = 0 for x > 0. What happens to y1 and y2 when
x approaches zero?
2. Solve 2x2 y 4xy + 6y = 0 for x > 0. What happens to y1 and y2
when x approaches zero?
3. Solve x2 y 3xy + 4y = 0 for x > 0. What happ
Worksheet 1: First order equations
For the following equations, determine whether they are linear or nonlinear
and then solve each of them.
1. xy + 2y = sin x.
The equation is equivalent to
2
sin x
y=
x
x
y +
for which the integrating factor is
2
dx
x
(x)
MATH 270: Dierential Equations
Final Exam
May 1, 2013
Please write clearly and put a box around your nal answer. Cross out any unwanted
material.
1. [10 points] A rst order equation of the form
y + P (x)y = Q(x)y n
(1)
with n = 0 or 1 is called the Bernou
Worksheet 5 Solutions
Consider the harmonic oscillator with an impulsive force y + y = (t 1).
The initial conditions are y(0) = 0 and y (0) = 0.
Solve this problem using the Laplace transform.
The transformed equation is s2 Y + Y = es which gives Y = es
Worksheet 6 solutions
1. Solve x2 y + 4xy + 2y = 0 for x > 0. What happens to y1 and y2 when
x approaches zero?
Substituting y = xr into the equation gives r 2 + 3r + 2 = 0 whose
roots are r = 1 and r = 2. Therefore the general solution is y =
c1 y1 + c2
MATH 270: Dierential Equations
Test 2
March 20, 2013
Please write clearly and put a box around your nal answer. Cross out any unwanted
material.
1. [10 points] Solve the system x = Ax with the following matrices A given below. Classify the
type of xed poi
MATH 270: Dierential Equations
Test 1
February 20, 2013
Please write clearly and put a box around your nal answer. Cross out any unwanted
material.
1. [5 points] Consider the logistic equation
dy
= (1 y) y
dt
that describes the time-evolution of a populat