MATH225, Spring 2026
Worksheet 1 (Review)
Name:
Due: Friday, January 16th
For full credit, you must show all work and box answers.
1. Find the roots of the following polynomials. (Solve for r.)
1
(a) − r − 54 = 0
6
(b) 6r2 + 54 = 0
(c) 6r2 + 5r = 4
(d) 3r3 − 15r2 + 12r = 0
(e) r2 + 6r + 13 = 0
2. Use exponential and logarithmic rules to simplify the following.
(a)
−3te−3t − e−3t
e4t
(b) e−2 ln(40 − t)
(c) e
3x +
1
ln(9)
2
3. Solve the equation
1
1
ln(y) − ln(3 − y) = 8 for y.
3
3
4. Verify the given y satisfies the given differential equation.
3
(a) y = − cos(5x) − sin(3x),
8
(b) y = −e2t + 3te2t ,
y ′′ + 9y = 6 cos(5x)
y ′′ − 4y ′ + 4y = 0
dP
=P
5. Given the differential equation
dt
P
1−
8
(a) Verify P = 3et is not a solution (does not satisfy the differential equation).
(b) Verify P = 8 is a solution (satisfies the differential equation).
6. Evaluate the following definite integrals.
Z e3
(ln x)3
(a)
dx
x
e
Z 1
(b)
0
e4t (t2 + 3)
dt
3et
Z 9 √x
e
√ dx
(c)
x
1
7. Evaluate the following indefinite integrals.
Z
6
(a)
dt
5 − 3t
Z
1
dx
4x − 3x2
Z
3y
dy
(y − 2)(y 2 + 4)
(b)
(c)
Algebra and Calculus Review
In MATH225 you will need to use algebra, trigonometry, and calculus as tools to solve ordinary differential equations.
Based on prerequisites, it is expected that you are skilled in these areas of mathematics already. Small arithmetic and
calculus errors may be graded more intensely than what you may have been accustomed to in your previous math classes.
This paper is intended to help review these topics but it should NOT be viewed as an all-encompassing review.
Quadratic Formula:
2
If ar + br + c = 0, then r =
−b ±
√
b2 − 4ac
2a
Exponential Properties:
ex
y
ex+y = ex ey , ex−y = y , (ex ) = exy ,
e
e−y =
1
,
ey
e0 = 1
Natural Log Properties:
x
,
y
Note: x, y can be any positive real numbers.
ln(x) + ln(y) = ln(xy),
y ln(x) = ln (xy ) ,
ln(x) − ln(y) = ln
eln(x) = x,
ln(1) = 0
b2
b
Completing the Square: as2 + bs + c = a(s − h)2 + k, where h = − , and k = c −
2a
4a
Ex:
2
2
−2
−2
2
2
s − 2s + 5 = s − 2s +
+5−
= (s − 1)2 + 4
2
2
2
2 2
1
1
1
19
s2 + s + 5 = s2 + s +
+5−
= s+
+
2
2
2
4
Partial Fraction Decomposition:
Ex:
s2 + s + 4
s(s − 1)(s + 2)
s2 + s + 4
s(s − 1)(s + 2)2
s2 + s + 4
s(s − 1)(s2 + 2)
=
=
=
A
B
C
+
+
s
s−1 s+2
B
C
D
A
+
+
+
s
s − 1 s + 2 (s + 2)2
A
B
Cs + D
+
+ 2
s
s−1
s +2
u-substitution:
Ex:
Z
xe
Z
Integration by Parts:
x2
Z
dx
=
Let u =
Z
u dv = uv − v du
1
1 2
1 u
e du = eu + C = ex + C
2
2
2
2
x , du = 2xdx
Ex:
Z
tet dt
= tet −
Let u = t,
du = dt,
Z
et dt = tet − et + C
dv = et dt
v = et