−
−
−
3. t2
−
1
?
?
?
?
ENGR 213 ASSIGNMENT
ASSIGNMENT
1 STUDY NOTES CONCORDIA UNIVERSITY
1. (1 point) It can be helpful to classify a differential equa-
tion, so that we can predict the techniques that might help us to
find a function which solves the equation. Two classifications
are the order of the equation – (what is the highest number of
derivatives involved) and whether or not the equation is linear
. Linearity is important because the structure of the the family
of solutions to a linear equation is fairly simple. Linear
equations can usually be solved completely and explicitly.
Determine whether or not each equation is linear:
1.
y
′′
−
y
+
t
2
=
0
2.
y
′′
y
+
y
2
=
0
d
2
y dy
+ + 2y
sin t
(a) The order of this differential equation is
.
(b) The equation is [Choose/Linear/Nonlinear].
5. (1 point) Find all values of k for which the function
y
=
sin
(
kt
)
satisfies
the
differential
equation
y
′′
+
7y
=
0.
Sep-
arate your answers by commas. Hint: There are more than
2 values of k
6. (1 point) Match the following differential equations with
their solutions.
The symbols A, B, C in the solutions stand for arbitrary con-
stants.
You must get all of the answers correct to receive credit.
dt2
4. d
2
y
dt
sin t y
=
sin t
d
2
y
dt2
+
( + )
=1. dx2 + 25y = 0
2. (1 point) In problems below, (a) identify the independent
variable and the dependent variable of each equation (use ’t’
for the independent variable if an independent variable is not
given explicitly); (b) give the order of each differential
equation (enter ’1’ for first order, ’2’ for second order and so
on; do not include the quotes); and (c) state whether the
equation is linear or nonlinear. If your answer to (c) is
nonlinear, make sure that you can explain why this is true.
equation
y
′
=
y
x
2
xy
′
=
2y
x
′′
+
5x
=
e
−
x
3. (1 point) Determine the order of the given differential
equation and state whether the equation is linear or nonlinear.
(
sin
θ
)
y
′′′
−
(
cos
θ
)
y
′
=
9
(a) The order of this differential equation is .
(b) The equation is [Choose/Linear/Nonlinear].
2. dy =
− 2 xy
dx x25y2
d
2
y dy
dx2 + 6 dx + 9y = 0
4. dy = 10xy
dx
5. dy + 9x2y = 9x2
dx
A.
3yx
2
5y3 = C
B. y = A cos(5x) + B sin(5x)
C.
y = Ae
−3x
+ Bxe
−3x
D. y = Ae5x2
(c)
linEe.ary/
n
=
onClein
−
e
3
a
x
r
3
+
1
[?/linear/nonlinear]
[?/l7in.
e(a1r/npoinlitn)
eWarh]ich
of
the
following
functions
are
solutions
[o?f/ltihneeadri/fnfeornelnintieaalre]quation
y
′′
−
10y
′
+
25y
=
0?
•
A. y(x) = e
−5x
•
B. y(x) = 5xe
−5x
•C. y(x) = xe5x
•D. y(x) = e5x
•E. y(x) = 0
5x•F. y(x) = x2e
•
G.
y
(
x
)
=
5
x
8. (1 point) If y = e3t is a solution to the differential equation
d
2
y dy
4. (1 point) Determine the order of the given differential
equation and state whether the equation is linear or nonlinear.
d
2
u du
3.
(a) independent (a) dependent (b) order
