Differential Equations

Differential equation-is an equation involving

derivatives
• Ordinary DE (ODE)- one independent
variable

y "' 2y " 3xy ' 5y  7  0

• Partial DE(PDE)- two or more independent

variables
x y
u u u 2 t 2
u
z  
 Differential Equations
• Order-is the order of the highest order
derivative of the unknown function explicitly
appearing in the equation.
y "' 2y " 3xy ' 5y  7  0

(y " 0.6y ')3  0.57y sin(0.16x)  0

 Solutions
Solution is any function (or formula for a function)
that satisfies the equation.

General solution -a formula that describes all

solutions to the equation.

Specific solution -a formula that describes a

solution at a boundary
- Initial Value Problem (IVP)
- Boundary Value Problem
Separable ODe

dy
dx  f
(x)g(y )
 Linear ODE

dy
dx  P(x)y  Q(x)

y  u   u Q(x)dx 
C

 P(x )dx
u e
 Homogenous ODE

dy y
dx  f  x 
substitution: y  vx
 Bernoulli ODE

dy n
dx  P(x)y  Q(x)y

y 1n  u  (1 n) u  Q(x)dx 

C

(1n) P( x )dx
u e
Exact Equations

Mdx  Ndy 
0
M N

y x

 Mx   Ny C
 Inexact Equations

Mdx  Ndy  0
M N

y x
N  M
y
x
For Integrating factor: let Q( x) 
M
Q( x )dx
u  e
Solution: solve
(u  M )dx  (u  N)dy 
0
 PROBLEM 1
Solve y '  2xy 2

1 1
A. y  B. y 
x2  C x
1 C 1
C. y   D. y  
x2  C x2  C
 PROBLEM 2
Solve y '  2xy 2

1 1
A. y  B. y 
x2  C x
1 C 1
C. y   D. y  
x2  C x2  C
 PROBLEM 3
Solve the ODE:
(x 5  3y )dx 
xdy  0.

B. y  2x 5
 cx 3
A. 2y  x 53  cx 35
C. 2y  x  cx
3 5
D. y  2x  cx
 PROBLEM 4
Solve the D.E.: 2(2x 2  y 2 )dx  xydy  0

A. x 4  C(4x 2  y
2
)
B. x 2  C(2x 2  y 2 )
C. x 2  C(2x 2  y
2
)
D. x 4  C(4x 2  y 2 )
 PROBLEM 5
Solve the equation xydx − x + 2y 2dy
= 0.
A. eX / y = c y 3 xy
C. eX / y = c y 3 (x + y)
B. eX / y = c y 3 x − y
D. eX / y = c y 4 x − 1
 PROBLEM 6
Solve the differential equation,

(6x  y 2 )dx  y(2x  3y )dy 

0
A. x2  y 3  C
B. 3x 2  y 3  xy  C
2

C. 2x 2  xy 3  y  C
2
 Linear Second-Order and Higher-
Order ODEs

General Form: y " p(x)y ' q(x)y  r

Homogenous Form: (x)
General Solution: y " p(x)y ' q(x)y  0
y  C1y1  C2 y 2  y p
where y1 and y2 are linearly independent solutions
 Solution to Second-Order ODEs
• Homogenous Solution, yh

• Particular Solution, yp
 Homogenous Solution, yh
If p(x), q(x) are constants,

y " ay ' by  r (x)

characteristic homogenous equation: (D2  aD  b)y 

0
a  a2  4b
characteristic solution: D
2
 Value of D Homogenous solution
DϵR
(one real root) y  C1eDx

D = m + ni
y  emx C1 sinnx  C2 cosnx 
D = R, R
(repeated roots) y  C1eRx  C2 xeRx

D = R, R, R
(repeated roots) y  C1eRx  C2 xeRx  C3 x 2eRx
 Particular Solution, yp
Method of Undetermined Coefficients
R(x) Yp Form
k (constant) yp  A

Polynomial, P(x) y p  Ax n  Bx n 1  ..  k

y p  (Ax n  Bx n 1  ..  k )sin mx
P(x)sin(mx)
+ (A ' x n  B ' x n 1  ..  k ')cos
mx

emxP(x) y p  e mx (Ax n  Bx n 1  ..  k )
 PROBLEM 7
Solve the D.E.: y " 7y ' 12y  5e 2 x

A. y  Ae3x  Be 4x  41 e 2x
B. y  Ae3x  Be 4x  61 e 2x
C. y  Ae3x Be4x 
4 1

D. y  Ae3x eBe2x4x 
6 1

e 2x
 PROBLEM 8
Solve the D.E.: y " 2y ' 8y  3e 2 x

A. y  Ae2x Be4x 4 1

B. y  Ae 2x
e 2x2 1 xe 2x

C. y  Ae 2 x  Be 4 x 21 xe 2x
Be4x 1 2x
D. y  Ae2x Be 4x  4 xe
 PROBLEM 9
Solve the D.E.: y " 4y ' 3y 
5
A. y  Ae x Be3x 5/3

B. y  Ae x 5/2
Be3x3x
C. y  Ae x  Be 5/2
D. y  Ae x  Be3 x  15 / 2
 PROBLEM 10
Solve the D.E.: y " 3y ' 2y  sin
2x
1
A. y   10 (3 cos 2x  2 sin 2x)

1
B. y   20 (3 cos 2x  sin 2x)

1
C. y   10 (cos 2x  sin 2x)

1
D. y  20 (2cos 2x  sin 2x)
 PROBLEM 11
Solve the particular solution of the D.E.:
y " 4y ' 7y  3 cos 5x
3
A. y p  
424 4 sin5x  5 cos 2x

3
B. y p  181 3 sin5x  5 cos 2x
3
C. y p  362 10 sin5x  9 cos 2x

3
D. y p  181 5 sin5x  9 sin 2x


 PROBLEM 12
Solve the D.E.: y " 2y ' y  2e  x sin3x

A. y  c1ex  c 2 xe x 91 e x sin(3x) + 3e

1 x
cos(3x)

B. y  c1ex  c 2 xe x 92 e x sin(3x)

C. y  c1e x
 c2 xe x 92 e x sin(3x) - 2 e  x cos(3x)
3

59 e x sin(3x)
D. y  c1ex  c 2 xe x
 PROBLEM 13
Determine the particular solution of y”’ – 8y =
4e^2x:

A. Yp = (1/2)xe^2x
B. Yp = (1/3)xe^2x
C. Yp = (1/4)xe^2x
D. Yp = (1/6)xe^2x
 PROBLEM 14
Determine the particular solution of y” – 10y’ +
24y = 6cos(3x):

A. Yp = (-2/25)(2sin(3x) – cos(3x))
B. Yp = (4/25)(sin(3x) – 2cos(3x))
C. Yp = (-1/25)(4sin(3x) +cos(3x))
D. Yp = (2/25)(4sin(3x) + cos(3x))
 PROBLEM 15
Determine the particular solution of y” +7y’
+12y = 8e^-xcos(2x):

A. Yp = (4/13)e^-x(6sin(2x) - cos(2x))
B. Yp = (5/13)e^-x(5sin(2x) - 2cos(2x))
C. Yp = (2/13)e^-x(4sin(2x) + cos(2x))
D. Yp = (2/13)e^-x(5sin(2x) + cos(2x))
 APPLICATIONS OF ODEs
• Newton’s Law of Cooling( or Heating)
• Unsteady-state Mass Balances
• Population Growth
• Nuclear Decay
 Nuclear Decay 13
Supposing that after 12 years, only 3.5% of the
original sample of a radioactive element
remains. Determine the half life:

A. 3.45 y
B. 1.56 y
C. 2.48 y
D. 2.16 y
 Nuclear Decay 14
Supposing that the half life of a radioisotope is
3.45 minutes. How long will it take for a 576
gram sample to disintegrate 570 grams?

A. 22.7 min
B. 14.5 min
C. 35.8 min
D. 10.3 min
 Nuclear Decay 15
A certain radioactive substance has a half – life
of 50 hours. How long will it take for 95% of the
substance to dissipate?

A. 255 hours C. 217 hours

B. 147 hours D. 342 hours
 Nuclear Decay 16
Dramstadtium isotope has a half-life of 567
days. Supposing that a sample has 78.56% less
radioactive Drmastadtium than a freshly
synthesized isotope. How old is the sample?
a. 3.5 years c. 6.3 years
b. 3.21 years d. 2.45
years
 Mixing Problem 17
A 500 gal tank initially contains 4 lb of salt.
Supposing that solution containing 2 lb/gal of salt
enters the tank at 10 gal/min and solution exits at
10 gal/min. The amount of saltin the tank after
30 minutes will be:

A. 233 lbs
B. 453 lbs
C. 567 lbs
D. 683 lbs
 Mixing Problem 18
A brine solution containing 1 gram/Li of salt enters a
1000-Liter tank initially containing 40 grams of salt in 500
Li of solution. The entering stream is at 10 Li/min and
the exiting stream is at 5 Li/min. Determine the amount
of salt in the tank when it is full ?

A. 1840 g
B. 2350 g
C. 770 g
D. 333 g
 Mixing Problem 19
Supposing a container with a volume of 450 ft3 initially
containing 20lb of sugar. A solution is entering this
container and it contains 0.5 lb/ft3 of solution. The inlet
and outlet solutions are both at 50 ft3/h. Determine the
time at which the concentration inside the container is
0.49 lb/ft3:

A. 1 day, 3 hours
B. 1 day, 10 hours
C. 2 days, 4 hours
D. 2 days, 14 hours
 Mixing Problem 20
A brine solution initially containing 300 lbs of salt 2000
gallons of solution is diluted using pure water entering at
a rate of 100 gal/min. The exiting stream is at 80 gal/min.
If the capacity of the tank is 4000 gallons, determine the
time at which the amount of salt is lowered to 50 lbs:

A. 12.3 min
B. 23.6 min
C. 3.45 min
D. 56 min
 Population Dynamics 21
The population in Timbuktu doubles every 12
years. How long would it take for the population
to triple?

A. 25 years
B. 19 years
C. 21 years
D. 27 years
 Population Dynamics 22
A population of insects in a region will grow at a rate that
is proportional to their current population. In the
absence of any outside factors the population will triple
in two weeks time. On any given day there is a net
migration into the area of 15 insects and 16 are eaten by
the local bird population and 7 die of natural causes. If
there are initially 100 insects in the area will the
population survive? If not, when do they die out?

A. 9.5 weeks C. 7.2 weeks

B. 2.3 weeks D. 10.8 weeks
 Newton’s Law of Cooling (or Heating)
23
A thermometer reading 75 degree Fahrenheit is
taken out where the temperature is 20 degree
Fahrenheit. The thermometer reading is 30 degree
Fahrenheit 4 minutes later. Find the time (in
minutes) taken for the reading to drop from 75
degree Fahrenheit to within half degree of the air
temperature.

A. 11 minutes C. 7 minutes
B. 20 minutes D. 16 minutes
 Newton’s Law of Cooling (or Heating)
24
A dead body was found in a lake at 13⁰C, at 5:30
PM. The body temperature at this time found
to be 17.6 ⁰C. It is immediately taken to a
nearby room at 24⁰C such that at 6:30 PM, the
temperature of the body is 20.4⁰C. Approximate
the time of death of this body:

A. 3:20 PM C. 3:50 PM
B. 2:35 PM D. 1:37 PM
 Newton’s Law of Cooling (or Heating)
25
At 1:00 PM, a thermometer reading 70 degF is
taken outdoors where the air temp is -10 degF.
At 1:02 PM, the reading becomes 26 degF. At
1:05 PM, the thermometer is taken back
indoors, where the temp is fixed at 70 degF.
What is the temperature reading at 1:09 PM?
A. 55 degF C. 57 degF
B. 56 degF D. 58 degF
 Newton’s Law of Cooling (or Heating)
16
A thermometer reading 18oC is brought into a
room where the temperature is 70oC; 1 minute
later the thermometer reading is 31 oC.
Determine the thermometer reading 5 minutes
after it is brought into the room.
A. 62.33 oC C. 56.55 oC
B. 58.99 oC D. 57.66 oC
 Newton’s Law of Cooling (or Heating)
27
A metal is heated up to a temperature of
500degC. It is then exposed to a temperature of
38degC. After 2 min, the temperature of the
metal becomes 190 degC. When will the
temperature be 100 degC?
a. 3.61 min b. 4.21
c. 3.44 min min
d. 5.22 min