Differential Equations Quiz
19 questions · no time limit
0:00
Finish quiz
FIRST-ORDER DIFFERENTIAL EQUATIONS
Question 1
of 19
Flag for review
No time limit
Current
Answered
Flagged
Not seen
Consider the equation
d
y
d
x
=
e
x
+
y
−
e
x
−
y
e
x
+
y
+
e
x
−
y
\frac{dy}{dx} = \frac{e^{x+y} - e^{x-y}}{e^{x+y} + e^{x-y}}
d
x
d
y
=
e
x
+
y
+
e
x
−
y
e
x
+
y
−
e
x
−
y
. To determine if this equation is separable, which analysis is most appropriate?
A
Factor out
e
x
e^x
e
x
from both numerator and denominator to obtain
d
y
d
x
=
e
y
−
e
−
y
e
y
+
e
−
y
=
tanh
(
y
)
\frac{dy}{dx} = \frac{e^y - e^{-y}}{e^y + e^{-y}} = \tanh(y)
d
x
d
y
=
e
y
+
e
−
y
e
y
−
e
−
y
=
tanh
(
y
)
, which is separable
B
Use the substitution
u
=
x
+
y
u = x + y
u
=
x
+
y
and
v
=
x
−
y
v = x - y
v
=
x
−
y
to transform the equation into a system that can be analyzed for separability
C
Recognize that
e
x
+
y
−
e
x
−
y
=
e
x
(
e
y
−
e
−
y
)
e^{x+y} - e^{x-y} = e^x(e^y - e^{-y})
e
x
+
y
−
e
x
−
y
=
e
x
(
e
y
−
e
−
y
)
and
e
x
+
y
+
e
x
−
y
=
e
x
(
e
y
+
e
−
y
)
e^{x+y} + e^{x-y} = e^x(e^y + e^{-y})
e
x
+
y
+
e
x
−
y
=
e
x
(
e
y
+
e
−
y
)
, so the equation separates immediately
D
The equation cannot be separated because the exponential terms create non-linear coupling between
x
x
x
and
y
y
y
that cannot be factored into independent functions
YOUR PROGRESS
19
QUESTIONS
Ready when you are
Answer in any order. Your progress shows here as you go.
QUESTIONS
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
KEYBOARD
Pick an answer
A
B
C
D
Next question
↵
→
Previous
←
Flag for review
F