One of life’s most painful moments comes when we must admit that we didn’t do our homework, that we are prepared. – Merlin Olsen
MATH 322
HAND-IN HW 1
Prof. B. M. Patiak
Name: XXXXXXXXXXDue: 1/25/2024
You must use the space provided for your work. You can use extra paper (must be computer paper) if the space
provided is NOT enough. All work should be written in PENCIL only. Write legibly and avoid erasures.
A. State the order of the given differential equation. Also, determine if the differential equation is linear or non-linear. If
the is non-linear, state your reason
iefly by identifying the non-linear term.
B. Verify that the indicated function is an explicit solution of the given differential equation.
5. 2?′ = ?3 cos ? , ? =
1
√1 − sin ?
One of life’s most painful moments comes when we must admit that we didn’t do our homework, that we are prepared. – Merlin Olsen
B. Find the particular solution of the following IVP. Then identify the largest interval where the solution is valid.
6. ?′ + 2??2 = 0, ?(−2) =
1
2
; ? =
1
?2 + ?
C. Find the particular solution of the following IVP.
7. ?′′ + ? = 0, ? (
?
4
) = √2, ?′ (
?
4
) = 2√2; ?(?) = ?1 cos(?) + ?2 sin(?)
One of life’s most painful moments comes when we must admit that we didn’t do our homework, that we are prepared. – Merlin Olsen
D. For the following differential equations, construct the slope field for [−3, 3] and sketch the solution curve at the
indicated points.
One of life’s most painful moments comes when we must admit that we didn’t do our homework, that we are prepared. – Merlin Olsen
10. Consider the autonomous differential equation
??
??
= ?(? − 1)(? + 2)
a) Identify all equili
ia of the system. Equili
ium solutions are when
??
??
= 0
) Draw the direction field for this system on −3 ≤ ? ≤ 3 and ? > 0. Include sufficient amount of integral curves to give
sense of the dynamics.
c) Determine how the solutions behave as ? → ∞ and if this behavior depends on the value of ?(0). Describe this
dependency.
One of life’s most painful moments comes when we must admit that we didn’t do our homework, that we are prepared. – Merlin Olsen
11. Consider the first order autonomous equation: ?′ = ?(2 − ?)(? + 3)2
a) Find all equili
ium solutions. Draw the direction or slope fields.
) If ?(2) = −3, what is ?(60)?
c) If ?(−6) = 0.3333, what is lim
?→∞
?(?) =