Activity 1.2.2: Skeleton Scavenger Hunt
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.
Non-homegeneous linear ODE, method of variation of parameters
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
1. For any piecewise continuous function f (x), the Legendre expansion is
2n + 1 1
f (x) =
an Pn (x),
f (x)Pn (x) dx; x [1, 1]
(i) We can use the above formula. Alternately, use 1 = P0 , x = P1 , x2 = (1+2P2
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.
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
1.T The equation y 00 + y 0 xy = 0 has a power series solution of the form y =
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
1. (a) Using separation of variables, we get
u(x, t) =
an exp(2n t) sin(nx/2),
n = n
Initial condition gives
an sin(nx/2) = sin
+ 3 sin
Using Fourier methods or by inspection we find a1 = 1, a5 = 3 and rest of th
Legendre Equation, Legendre Polynomial
This equation arises in many problems in physics, specially in boundary value problems
(1 x2 )y 00 2xy 0 + ( + 1)y = 0,
where is a constant.
We write this
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 +
1. (i) x = 0, x = 1 regular, x = 1 irregular (ii) x = 0, x = 1 regular
2. In all the problems, substitute y =
an xn+r , a0 6= 0, to get
(a) [9r(r 1) + 2]a0 = 0, [9r(r + 1) + 2]a1 = 0 and
, n 2.
9(n + r)(n +
Laplace Transform, inverse Laplace Transform, Existence and Properties of Laplace
Differential equations, whether ordinary or partial, describe the ways certain quantities
of interest vary over time. These
Picards existence and uniquness theorem, Picards iteration
Existence and uniqueness theorem
Here we concentrate on the solution of the first order IVP
y 0 = f (x, y),
y(x0 ) = y0
We are interested in the following questions:
Homogeneous Euler-Cauchy equation
If the ODE is of the form
ax2 y 00 + bxy 00 + cy = 0,
where a, b and c are constants; then (1) is called homogeneous Euler-Cauchy equation.
Two linearly independent solut
3.3.1.A Med History 1.docx
Activity 3.3.1: Medical History Visit #1
ARTUR SETYAN 3.3.4 Asthma Action plan
Asthma Action Plan
Personal best peak flow:
Name: Melissa Martin
Doctor name: Dr. Setyan
Doctor phone: 818-123-4568
Medicine: Accolate, Proair
How much: 10 milligram tabl
Activity 3.3.1: Medical History Visit #1
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
Tongue: contains rough papil
Human Body Systems
Voluntary activation of the Quadriceps muscle
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
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
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 +
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 /
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
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:
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
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:
1. It is clear that f (0) = g(0) = 0. Define the odd extension of f and g by
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),