1. MATH2860U Midterm October, 2008 2
1. (4 marks each; total 12 marks) Answer each question in the space provided. You
MUST show all of your work.
a) Consider the following linear first-order differential equation:
xy′ − 3 y = sin x 3 . What is the appropriate integrating factor to help solve this
equation? (Just find the integrating factor, but do NOT actually solve the
equation).
m
b) Find all values of m so that x is a solution of the differential equation
x 2 y′′ + 2 xy′ − 2 y = 0
c) Find the general solution of the differential equation
( D + 1)( D 2 + 9)( D 2 − 16) 2 y = 0 .

2. MATH2860U Midterm October, 2008 3
2. (7 marks) Find the general solution of the differential equation
⎛ 1 ⎞ dy
(4 x 3 y 2 − e − x ) + ⎜ 2 x 4 y + ⎟ = 0
⎜
⎝ y ⎟ dx
⎠

3. MATH2860U Midterm October, 2008 4
3. (Total 12 marks) Write your final answer on the line provided. You do NOT have to
show your work, but it is suggested for parts f and g (for part marks in case the final
answer is wrong). For all other parts, ONLY the final answer will be marked.
a) (2 marks) Suppose that we have a radioactive series in which element A decays
to element B then C with rate constants k1 , k 2 as shown below ( k1 , k 2 positive).
dA
The differential equation governing the amount of element A is = − k1 A .
dt
What are the differential equations governing the amount of element B and
element C?
Differential Equation for Element B: _________________________________
Differential Equation for Element C: _________________________________
b) (1 mark) Determine whether the functions y1 = x 2 and y 2 = x 2 + 1 are linearly
dependent or linearly independent. (Answer: “Dependent” or “Independent”)
Answer : ________________________________________
c) (1 mark) Which of the following statements describes the differential equation
x 3 y ′′ + (cos x) y − x 2 e x = 0 ?
i. Linear and homogeneous
ii. Linear and nonhomogeneous
iii. Nonlinear
Answer: _________________________________
d) (3 marks) For the equation y′′ + y′ − 12 y = 6 x 2 + 2e −4 x , if it is known
−4 x
that yc = c1e + c2 e , what is the appropriate form to be assumed for the
3x
particular solution, if solving using the method of undetermined coefficients? You do
NOT have to solve for the values of the constants A, B, C, etc.
y p is of the form: _________________________________

4. MATH2860U Midterm October, 2008 5
e) (1 mark) Without attempting to solve the ODE, determine whether the ODE
y′ = x 4 − y 2 is guaranteed to have a unique solution through the point (2,0).
Answer “True” or “False”.
Answer: _________________________________
f) (2 marks) Suppose an LRC circuit is critically damped. Increasing the resistance
of the resistor will result in what type of motion?
i) underdamped motion.
ii) overdamped motion.
iii) pure resonant motion
iv) both a and c
Answer: _________________________________
g) (2 marks) Consider the differential equation y′ = y x + x subject to the
2
initial condition y(3) = 7. Use Euler’s Method with a step size of h = 0.2 to
approximate y(3.2).
Answer: _________________________________

5. MATH2860U Midterm October, 2008 6
4. (9 marks) A horizontal spring with a mass of 2 kg has damping constant 8, and a
force of 6 N is required to keep the spring stretched 0.5 m beyond its natural length. The
spring is stretched 1m beyond its natural length (i.e. to the right) and then released with
zero velocity. Find the position x(t) of the mass at any time t. Note: There is no
external force F(t) applied to the spring.

6. MATH2860U Midterm October, 2008 7
5. (10 marks total) Use variation of parameters to obtain the particular solution of the
given differential equation y′′ + y′ − 12 y = 8e NOTE: You must use variation of
2x
parameters; other methods will NOT receive any marks. Hint: You may use the fact that
yc = c1e −4 x + c2e3 x for this differential equation.