![Concordia University
EMAT 213 - Final Exam
Instructors: Bertola, David, Frank, Jones, Helfgott, Keviczky, Kokotov, Stern
Course Examiner: M. Bertola
Date: December 2005.
Time allowed: 3 hours.
Directions: non graphing calculators are allowed.
Note: Marked out of 100 points
[30 pts] Problem 1
Solve the following first order ODEs
2
(a) 2x cos(x2 + y 2 ) − 4x3 y dx + 2y cos(x2 + y 2 ) − x4 + 2yey dy = 0
ey+5x+3
(b) y = −5 [Hint: with a substitution u = Ax + By + C it becomes separable]
y + 5x + 3
x2
(c) x3 + 1 y = 2 .
y
[20 pts] Problem 2
Solve the following linear ODEs by finding the general solution.
(a) y (7) − 6y (6) + 20y (5) − 56y (4) + 112y (3) − 160y + 192y − 128y = x ;
Note : m7 − 6m6 + 20m5 − 56m4 + 112m3 − 160m2 + 192m − 128 = (m − 2)3 (m2 + 4)2
(b) x2 y − 2y = x2 , (x > 0)
Note : variation of parameters [put it in standard form!].
[10 pts] Problem 3
Solve the following nonlinear ODE of second order
(y + 2)y = (y )2
using the substitution y = u(y), y = u du (variable x is missing).
dy
[10 pts] Problem 4
Find the general solution as a power series centered at the regular point x = 0 of the differential equation
y + x2 y = 0
Give only the first six terms of the series (namely up to degree 5).
1](https://image.slidesharecdn.com/emat213finalfall2005-110314122123-phpapp01/85/Emat-213-final-fall-2005-1-320.jpg)
![[10 pts] Problem 5
Solve the following nonhomogeneous linear system of ODEs
dx = x + 2y + t
dt
dy
=x+1
dt
by finding Xc (complementary solution), Xp (particular solution).
[10 pts] Problem 6
A cake is removed from an oven at a temperature of 150 degrees (Celsius) and left to cool down on a
countertop in a room at a temperature of 20 degrees. After 2 minutes the temperature of the cake is
120 degrees. After how many minutes (approximately) the cake’s temperature will have dropped to 40
degrees?
[15 pts] Problem 7
[10 pts] A spring of constant k = 2N/m is immersed in a viscous fluid. A mass of one Kilogram is attached
to the loose end and it is released half a meter below the equilibrium position without any initial velocity.
Find the motion of the mass knowing that there is a damping coefficient equal to 2 N × s/m.
[5pts] The viscous fluid is removed. The mass is now subject to an external force
Fext = 0.001 sin(γt) Newtons.
How should we choose γ such as to break the spring in the long run? Explain.
2](https://image.slidesharecdn.com/emat213finalfall2005-110314122123-phpapp01/85/Emat-213-final-fall-2005-2-320.jpg)
Emat 213 final fall 2005
- 1. Concordia University EMAT 213 - Final Exam Instructors: Bertola, David, Frank, Jones, Helfgott, Keviczky, Kokotov, Stern Course Examiner: M. Bertola Date: December 2005. Time allowed: 3 hours. Directions: non graphing calculators are allowed. Note: Marked out of 100 points [30 pts] Problem 1 Solve the following first order ODEs 2 (a) 2x cos(x2 + y 2 ) − 4x3 y dx + 2y cos(x2 + y 2 ) − x4 + 2yey dy = 0 ey+5x+3 (b) y = −5 [Hint: with a substitution u = Ax + By + C it becomes separable] y + 5x + 3 x2 (c) x3 + 1 y = 2 . y [20 pts] Problem 2 Solve the following linear ODEs by finding the general solution. (a) y (7) − 6y (6) + 20y (5) − 56y (4) + 112y (3) − 160y + 192y − 128y = x ; Note : m7 − 6m6 + 20m5 − 56m4 + 112m3 − 160m2 + 192m − 128 = (m − 2)3 (m2 + 4)2 (b) x2 y − 2y = x2 , (x > 0) Note : variation of parameters [put it in standard form!]. [10 pts] Problem 3 Solve the following nonlinear ODE of second order (y + 2)y = (y )2 using the substitution y = u(y), y = u du (variable x is missing). dy [10 pts] Problem 4 Find the general solution as a power series centered at the regular point x = 0 of the differential equation y + x2 y = 0 Give only the first six terms of the series (namely up to degree 5). 1
- 2. [10 pts] Problem 5 Solve the following nonhomogeneous linear system of ODEs dx = x + 2y + t dt dy =x+1 dt by finding Xc (complementary solution), Xp (particular solution). [10 pts] Problem 6 A cake is removed from an oven at a temperature of 150 degrees (Celsius) and left to cool down on a countertop in a room at a temperature of 20 degrees. After 2 minutes the temperature of the cake is 120 degrees. After how many minutes (approximately) the cake’s temperature will have dropped to 40 degrees? [15 pts] Problem 7 [10 pts] A spring of constant k = 2N/m is immersed in a viscous fluid. A mass of one Kilogram is attached to the loose end and it is released half a meter below the equilibrium position without any initial velocity. Find the motion of the mass knowing that there is a damping coefficient equal to 2 N × s/m. [5pts] The viscous fluid is removed. The mass is now subject to an external force Fext = 0.001 sin(γt) Newtons. How should we choose γ such as to break the spring in the long run? Explain. 2