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HW 1 DIFF EQ SP 23
Dr. Elliott Hollifield
1. y = e3x and y = e4x are solutions of
y ′′ − 7y ′ + 12y = 0 .
Show that ae2x + be3x is also a solution for any a, b ∈ R
2. Show that xy = log y + C is a solution to
dy
y2
=
dx
1 − xy
by explicitly computing derivatives.
3. Show that ex
2
R x −t2
e dt is a solution to
0
y ′ = 2xy + 1
by explicitly computing derivatives.
4. For what values of the constant m will y = emx be a solution of the
differential equation
2y ′′′ + y ′′ − 5y ′ + 2y = 0 ?
5. Solve the differential Equations using separation of variables:
2
(a) yy ′ = ey sin x
dy
(b) dx
= x−xy
x2 +1
1
6. Find all values of m so that the function f (x) = emx is a solution to
y ′′ + y ′ − 6y = 0.
7. The constant solutions of the differential equation y ′ = 5y − y 2 are
y(x) = 0 and y(x) = 5.
(a) Find intervals on the y axis on which a non-constant solution is
increasing.
(b) Find intervals on the y axis on which any non-constant solution
is decreasing.
8. Solve the differential Equations and find the interval of validity for the
solutions
(a)
1
dy
= 6y 2 x where y(1) =
dx
25
(b)
y′ =
3×2 + 4x − 4
where y(1) = 3
2y − 4
2
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