Activity 1.2.2: Skeleton Scavenger Hunt
Introduction
Throughout the Human Body Systems course, you will explore the many functions of
the skeletal system. Bones, cartilage, ligaments, and tendons are all types of
connective tissue that support your frame.

S. Ghorai
1
Lecture X
Non-homegeneous linear ODE, method of variation of parameters
0.1
Method of variation of parameters
Again we concentrate on 2nd order equation but it can be applied to higher order ODE.
This has much more applicability than the metho

MTH203: Assignment-8:Solutions
1. For any piecewise continuous function f (x), the Legendre expansion is
Z
X
2n + 1 1
f (x) =
an Pn (x),
an =
f (x)Pn (x) dx; x [1, 1]
2
1
n=0
(i) We can use the above formula. Alternately, use 1 = P0 , x = P1 , x2 = (1+2P2

MTH203: Assignment-13:Solutions
1. Verification is easy.
2. (a) u(x, y) = x + y satisfies the Laplace equation and the boundary conditions. Hence,
by maximum principle, u(x, y) = x + y is the solution
(b) Similar to (a) and u(x, y) = xy is the solution.
3

MTH203: Assignment-2: Solutions
1. (i) Use transformation x = X + h, y = Y + k such that h + 2k + 1 = 2h + k 1 = 0.
Thus h = 1, k = 1 and the ODE becomes dY /dX = (X + 2Y )/(2X + Y ). Further
substitution of v = Y /X leads to separable form Xdv/dX = (1 v

MTH203: Assignment-6
1.T The equation y 00 + y 0 xy = 0 has a power series solution of the form y =
P
an x n .
(i) Find the power series solutions y1 (x) and y2 (x) such that y1 (0) = 1, y10 (0) = 0 and
y2 (0) = 0, y20 (0) = 1.
(ii) Find the radius of con

MTH203: Assignment-14
1. (a) Using separation of variables, we get
u(x, t) =
X
an exp(2n t) sin(nx/2),
n = n
n=1
Initial condition gives
X
an sin(nx/2) = sin
n=1
x
5x
+ 3 sin
2
2
Using Fourier methods or by inspection we find a1 = 1, a5 = 3 and rest of th

S. Ghorai
1
Lecture XIII
Legendre Equation, Legendre Polynomial
1
Legendre equation
This equation arises in many problems in physics, specially in boundary value problems
in spheres:
(1 x2 )y 00 2xy 0 + ( + 1)y = 0,
(1)
where is a constant.
We write this

MTH203:Assignment-10:Solutions
1. (i) z = xy + F (x2 + y 2 ) = p = y + 2xF 0 , q = x + 2yF 0 . Eliminating F 0 we get
py qx = y 2 x2 .
(ii) z = F (x/y) = p = F 0 /y, q = xF 0 /y 2 . Eliminating F 0 we get px + qy = 0.
2. (i) z = (x + a)(y + b) = p = (y +

MTH203: Assignment-7:Solutions
1. (i) x = 0, x = 1 regular, x = 1 irregular (ii) x = 0, x = 1 regular
2. In all the problems, substitute y =
P
n=0
an xn+r , a0 6= 0, to get
(a) [9r(r 1) + 2]a0 = 0, [9r(r + 1) + 2]a1 = 0 and
an =
9an2
, n 2.
9(n + r)(n +

S. Ghorai
1
Lecture XVII
Laplace Transform, inverse Laplace Transform, Existence and Properties of Laplace
Transform
1
Introduction
Differential equations, whether ordinary or partial, describe the ways certain quantities
of interest vary over time. These

S. Ghorai
1
Lecture V
Picards existence and uniquness theorem, Picards iteration
1
Existence and uniqueness theorem
Here we concentrate on the solution of the first order IVP
y 0 = f (x, y),
y(x0 ) = y0
(1)
We are interested in the following questions:
1.

S. Ghorai
1
Lecture XI
Euler-Cauchy Equation
1
Homogeneous Euler-Cauchy equation
If the ODE is of the form
ax2 y 00 + bxy 00 + cy = 0,
(1)
where a, b and c are constants; then (1) is called homogeneous Euler-Cauchy equation.
Two linearly independent solut

Artur Setyan
3.3.1.A Med History 1.docx
Activity 3.3.1: Medical History Visit #1
Patients Name:
Melissa Martin
Height:
52 inches
Blood Pressure:
110/72
Age:
11
Weight:
70 lbs.
Pulse:
75 bpm
Date:
February 5
Temperature:
98.3F
Respiration Rate:
22 bpm
Case

ARTUR SETYAN 3.3.4 Asthma Action plan
Asthma Action Plan
Personal best peak flow:
Important Info
Name: Melissa Martin
Date:
Doctor name: Dr. Setyan
Doctor phone: 818-123-4568
Exercise-Induced Flare-Up
Medicine: Accolate, Proair
How much: 10 milligram tabl

Artur Setyan
Activity 3.3.1: Medical History Visit #1
Patients Name:
Melissa Martin
Height:
52 inches
Blood Pressure:
110/72
Age:
11
Weight:
70 lbs.
Pulse:
75 bpm
Date:
February 5
Temperature:
98.3F
Respiration Rate:
22 bpm
Case History
Melissa is an 11 y

Mouth: oral cavity teeth tongue pharynx salivary glands.
Teeth: start mechanical digestion which is the physical breakdown of food into
smaller pieces. Teeth chew and grind the food particles in a process known as
mastication.
Tongue: contains rough papil

Artur Setyan
Milton Loango
Human Body Systems
3/7/17
Voluntary activation of the Quadriceps muscle
Kick 1
Kick 2
Kick 3
Kick 4
Kick 5
Average
Time of
muscle
contraction
(s)
1.590
1.62
1.13
1.11
1.01
1.292
Time of
Stimulus
0.215
0.99
0.52
0.57
0.52
0.563
t

Artur Setyan
4/5/17
Human Body systems
3.4.3.P Nephron.docx procedure questions and notes
o Calculate the GFR per day. Show your work. Express your
answer in liters (L) per day.
GFR= 125 mL every minute
One day= 1440 minutes
125 x 1440= 180,000 mL
180,000

S. Ghorai
1
Lecture I
Introduction, concept of solutions, application
Definition 1. A differential equation (DE) is a relation that contains a finite set of
functions and their derivatives with respect to one or more independent variables.
Definition 2. A

MTH203: Assignment-6:Solutions
P
P
1. (i) Substituting y = n=0 an xn into y 00 + xy 0 + y = 0, we get
n=0 [(n + 2)(n + 1)an+2 +
n
nan + an ]x = 0. Thus, an+2 = an /(n + 2). Iterating a2 = a0 /2, a4 = a0 /(2 4), a6 =
a0 /(2 4 6), and a3 = a1 /3, a5 = a1 /

MTH-102: Assignment-7:Solutions
1. Let on the contrary, u(x) has infinite number of zeros in [a, b]. It follows that there exists
x0 [a, b] and a sequence of zeros xn 6= x0 such that xn x0 . Since u(x) is continuous
and differentiable at x0 , we have
u(x0

MTH203: Assignment-7
1.D Expand the following functions in terms of Legendre polynomials over [1, 1]:
(
0 if 1 x < 0
(i) f (x) = x3 +x+1
(ii) f (x) =
(first three nonzero terms)
x if 0 x 1
2.T Locate and classify the singular points in the following:
(i)

MTH203: Assignment-4:Solutions
1. (i) Dependent variable y absent. Substitute y 0 = p = y 00 = dp/dx. Thus xp0 + p = p2 .
Solving p = 1/(1 ax) which on integrating again gives y = b ln(1 ax)/a.
(ii) Independent variable x absent. Substitute y 0 = p = y 00

MATH-102: Assignment-4
1.
Find the curve y = y(x) passing through origin for which y 00 = 2y 0 and the line y = x
is tangent at the origin.
2. Find the differential equation satisfied by each of the following two-parameter families
of plane curves:
(i) y

MTH203:Assignment-12:Solutions
1. It is clear that f (0) = g(0) = 0. Define the odd extension of f and g by
(
(
f (x)
x0
g(x)
x0
F (x) =
and G(x) =
f (x) x 0
g(x) x 0
Consider the IBVP
vtt = c2 vxx ,
x (, ), t > 0
v(x, 0) = F (x),
x (, )
vt (x, 0) = G(x),