MAT 275 TEST 1 PRACTICE
1.
Determine if each of the following equations is separable (Yes or No), and /or linear (Yes or No).
Record your answer in the following table. Do not attempt to solve the equations.
Equation
Separable
Linear
?
′
=
? + 1
??
?
′
=
??
? + 1
?
′
= cos (??)
?
′
− ?? = ?
3
2.
a) Write down a first order linear ODE whose solutions all approach
? = 1
b) Write down a first order linear ODE such that solutions other than
? = −3
all diverge from
? = −3
3.
Consider the ODE
?
′
= 3?
4
+ 3?
3
− 6?
a)
Determine all the equilibrium (constant) solutions and classify them as stable, unstable or semi-stable.
b)
For what values of
?
is
?
increasing?
4.
Which of the following differential equations best
represents the slope field at the right?
a)
?
′
= ? + ?
b)
?
′
= ? − ?
c)
?
′
= ? − ?
d)
?
′
= ??
e)
?
′
=
?
?
.
.
2

5.
Which
of the following differential equations best represents the slope field
below:
I.
?
′
= (1 − ?)?
II.
?
′
= ??
−?
III.
?
′
= ? − ?
6.
Consider the differential equation:
?
2
?
′′
− 3??
′
+ 5? = 2?
2
ln(?)
(a) What is the order of the differential equation?
(b)
Is the differential equation linear or nonlinear?
7.
Consider the differential equation:
−4? + ?
2
+ 2???
′
= 0
(a)
What is the order of the differential equation?
(b)
Is the Differential Equation linear or nonlinear?
.
.

9.
The general solution to a certain first order differential equation appears in implicit form
?
2
− sin(?) = 𝐶
.
Determine the value at
? =
𝜋
2
of the solution satisfying
?(0) = −1
.
10.
The general solution of the differential equation
??? = ???
is a family of (determine the correct choice)
a) circles
b) parabolas
c) hyperbolas
d) lines passing through the origin
11.
Use separation of variables to find the solution of the following Initial Value Problems.
Write your answer in explicit form and simplify as much as possible.
For each IVP determine the
interval in which the solution is defined.
a)
?
′
=
2?
?+1
,
?(1) = −2
b)
??
2
+ 3?
2
− ?
2
?