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 /

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

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 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

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

MTH-102: Assignment-7
1. (D) Let u(x) be any nontrivial solution of u00 + q(x)u = 0 on a closed interval [a, b].
Show that u(x) has at most a finite number of zeros in [a, b].
2. (D) Show that any nontrivial solution of u00 + q(x)u = 0, q(x) < 0 has at mo

MTH203: Assignment-8
1.D Let F (s) be the Laplace transform of f (t). Find the Laplace transform of f (at) (a > 0).
2.D Find the Laplace transforms:
(a) [t] (greatest integer function), (b) tm cosh bt (m non-negative integers),
(
t
sin 3t,
0 < t < ,
e
sin

S. Ghorai
1
Lecture VII
Second order linear ODE, fundamental solutions, reduction of order
A second order linear ODE can be written as
y 00 + p(x)y 0 + q(x)y = r(x),
x I,
(1)
where I is an interval. If r(x) = 0, x I, then (1) is a homogeneous 2nd order li

MTH203N: Assignment-I
1.D Classify each of the following differential equations as linear, nonlinear and specify the
order
(i) y 00 + y sin x = 0
(ii) y 00 + x sin y = 0 (iii) y 0 = 1 + y
(iv) y 00 + (y 0 )2 + y = x (v) y 00 + xy 0 = cos y 0 (vi) (xy 0 )0

MTH 102: Assignment-2
1. Find general solution of the following differential equations:
(i) (x + 2y + 1) (2x + y 1)y 0 = 0 (ii)(D) y 0 = (8x 2y + 1)2 /(4x y 1)2
2. Show that the following equations are exact and hence find their general solution:
2
2
(i)

MTH203: Assignment-9:Solutions
1.
L
Z
Z
ebt
1 (sb/a)t
f (t/a) =
e
f (t/a) dt =
e(asb) f ( ) d = F (as b)
a
a 0
0
2. (a)
L(sin at) =
s2
a
a
= L(et sin at) =
2
+a
(s 1)2 + a2
(b)
(2m)!
1
= L(t2m cosh bt) = L(ebt t2m + ebt t2m )
2m+1
s
2
(2m)!
1
1
=
+
2m+1

S. Ghorai
1
Lecture XVIII
Unit step function, Laplace Transform of Derivatives and Integration, Derivative and
Integration of Laplace Transforms
1
Unit step function ua(t)
Definition 1. The unit step function (or Heaviside function) ua (t) is defined
0,
t

MTH203:Assignment-10
1. In the following, by elimination of the arbitrary function F , find a first order p.d.e.
satisfied by z:
(i) z = xy + F (x2 + y 2 )
(ii) z = F (x/y)
2. In the following, by elimination of the arbitrary constants, find a first order

MTH203: Assignment-9:Solutions
1. (i) 0 leads to trivial solution. Thus, let = p2 > 0. Then y = c1 cos px + c2 sin px.
Using the BCs c1 = 0 and sin p + p cos p = 0 or p + tan p = 0. This has infinite number
of roots (plot the curves y = x and y = tan x).