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 /
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
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:
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
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
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
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),
0 < t < ,
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),
where I is an interval. If r(x) = 0, x I, then (1) is a homogeneous 2nd order li
1.D Classify each of the following differential equations as linear, nonlinear and specify the
(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:
f (t/a) =
f (t/a) dt =
e(asb) f ( ) d = F (as b)
L(sin at) =
= L(et sin at) =
(s 1)2 + a2
= L(t2m cosh bt) = L(ebt t2m + ebt t2m )
Unit step function, Laplace Transform of Derivatives and Integration, Derivative and
Integration of Laplace Transforms
Unit step function ua(t)
Definition 1. The unit step function (or Heaviside function) ua (t) is defined
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
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).