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Differential Operator - Mathematics - Exam, Exams of Mathematics

Chhattisgarh Swami Vivekanand Technical UniversityMathematics

This is the Exam of Mathematics which includes Four Roots, Quartic, Factorize the Quartic, Eulerian Representation, Lagrangian Position, Euler Lagrange Equations, Extrema, Functional etc. Key important points are: Differential Operator, Solution, Homogenous Equation, Wronskian, Construct, Second Solution, Initial Value Problem, Appropriate, Homogenous, Regular Singular Point

Typology: Exams

2012/2013

Uploaded on 02/23/2013

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1. The differential operator Lis defined by
Ly ≡y00 −2y0+1−2
x2y
in 1<x<∞.
(a) Show that y(x) = x2exis a solution to the homogenous equation Ly = 0 .
(b) Find a Wronskian for the equation Ly = 0 and hence construct a second solution to
the homogenous equation.
(c) Using a Green’s Function as appropriate for an initial value problem, solve
Ly =xex
with y(1) = y0(1) = 0 .
2. Consider the 2nd order homogenous ODE
xy00 + (1 −x)y0+ky = 0 .
(a) Show that it has a regular singular point at x= 0.
(b) Find one power series solution to the ODE about the point x= 0, which is finite at
x= 0.
(c) What is the radius of convergence of the series solution?
(d) Determine the values of the parameter k,=νsay, for which the series terminates as
a polynomial; call such solutions Lν(x).
(e) Rewrite the ODE in Sturm-Liouville form and hence deduce an orthogonality
relationship for the different polynomial solutions Lνand Lμ(x),ν6=μ.
M2M2 Differential Equations (2005) Page 2 of 4
pf3

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1. The differential operator L is defined by

Ly ≡ y′′^ − 2 y′^ +

1 − (^) x^22

y

in 1 < x < ∞. (a) Show that y(x) = x^2 ex^ is a solution to the homogenous equation Ly = 0. (b) Find a Wronskian for the equation Ly = 0 and hence construct a second solution to the homogenous equation. (c) Using a Green’s Function as appropriate for an initial value problem, solve Ly = xex with y(1) = y′(1) = 0.

2. Consider the 2nd order homogenous ODE

xy′′^ + (1 − x)y′^ + ky = 0. (a) Show that it has a regular singular point at x = 0. (b) Find one power series solution to the ODE about the point x = 0, which is finite at x = 0. (c) What is the radius of convergence of the series solution? (d) Determine the values of the parameter k, = ν say, for which the series terminates as a polynomial; call such solutions Lν (x). (e) Rewrite the ODE in Sturm-Liouville form and hence deduce an orthogonality relationship for the different polynomial solutions Lν and Lμ(x), ν 6 = μ.

3. Using the complex form of the Fourier series, show that for 0 < x < 2 π

ex^ = (^21) π^ (e^2 π^ − 1 )^

n=−∞

einx 1 − in. By considering a particular value of x, deduce that

π coth π = π (e

2 π (^) + 1) e^2 π^ − 1 = 1 + 2

∑^ ∞

n=

1 + n^2. Similarly, show that a second choice for x leads to the result

π cosech π = 1 + 2

∑^ ∞

n=

(−1)n 1 + n^2.

4. The Fourier Transform of the function f (x) is defined by

f̂ (k) =

−∞^ f^ (x)^ e

−ikx (^) dx.

State the inverse transform which defines f (x) in terms of f̂ (k). The function f (x) is defined as

f (x) =

{ (^) e−x (^) for x > 0 , −ex^ for x < 0. (a) Find f̂ (k). (b) Use the convolution theorem to find the function g(x) which satisfies ∫ (^) ∞ −∞

f (y)g(x − y)dy = xg(x)

and takes the value g(0) = 1/ 2.