Question

State whether the following statements are True or False. Justify your answer with the help of a short proof or a counter-example:

i)  Initial value problem:

\frac{dy}{dx}=\frac{y-1}{x}

y (0)=1 has a unique solution

ii)  The second order Runge-Kutta method when applied to IVP y′ = −100y,  y(0)=1 will produce stable results for  0<h<\frac{1}{50}.

iii) If Fourier cosine transform of  f(x ) is:

F_{c}(n)= \frac{cos(\frac{2n\pi }{3})}{(2n+1)^2}

where  0 ≤ x ≤ then:

f(x)=1+2\sum_{n=1 }^{\infty}\frac{cos(\frac{2n\pi }{3})}{(2n+1)}cosn \pi x.

For the differential equation x_{2}(x-4)^2y''(x)+3xy'(x)-(x-4)y=0,x=0 is a 

regular singular point and x=4,is an irregular singular point.

22 Mar 2023
Answer :
Word Count : 593

i) True:

We can write the given first-order differential equation as:

\frac{1}{y-1} dy = \frac{1}{x} dx

Integrating both sides, we get:

ln|y-1| = ln|x| + C

where C is the constant of integration. Solving for y, we get:

y = 1 + Ce^ln|x| = xC' + 1

where C' = e^C is another constant of integration. Using the initial condition y(0) = 1, we get C' = 1. Therefore, the unique solution to the given initial value problem is:

y = x + 1

ii) False:

The second order ______ ________ ______ ___ ____ _______ ________ _________.
___ _________ ________ ______ ______ __________ ___ ___.
_________ _______ _________ _______ ____ _________ ______.
______ _________ __________ ____ _________ ____ _____ ________ __________ ____.
________ __________ _____ _____ _______ ________ ___ ____ ___ _______ ____ __________.
________ _____ _________ _____ _________ ___ ______ _________ __________.
___ ___ ________ ________ _______ ___ __________ ________ _________ ____ _______ ___.
__________ _______ ________ ____ ___.
___ ______ ______ ______ _________ ______ _______ ______.
__________ ______ _________ _____ ______ _______ ___ ___ _______ _____ ________ _____.
___ _____ _____ _______ __________ ___ _____ ________ ___ _____ ____ _______.
____ __________ _______ ________ _____ _______ ________ ____.
_____ ________ _____ ______ __________ ______ ______.
_____ _________ ____ ____ ___ __________ _________ _______ _____ __________.
___ _____ ____ ______ _________ _______.
___ _____ __________ _________ __________ ______ _________ ________ __________.
______ ____ __________ _____ _________.
_____ ___ ____ ___ __________ ____ __________.
____ _____ _____ _________ ________.
________ ___ _____ _____ ____ _____ __________.
_______ _______ _______ ______ ____ ____ _____ _____ __________ _____ _____ __________.
______ __________ _____ ____ ________ _______.
____ __________ _______ __________ ____ _________ _____.
_______ _________ ____ ___ ____ _________ ___ __________ _________ _________ ______ ____.
________ __________ _______ ______ ______ ____ ___ _____ _________.
________ _______ ______ ___ ____ ________ __________ ____ _____ ________ ___.
_____ ________ _______ ______ ________ ____ _____.
__________ ______ ____ ______ ______ __________.
__________ ________ __________ ___ ____ _____ _________ ________ ___.
______ __________ _____ ______ _______ _______ ____ _____ ______.
_____ ________ ______ ___ __________ _______ _____.
_____ __________ _________ __________ _____ ______ __________.
____ ________ _________ __________ _______.
__________ ___ _____ ______ _________ ______.
_________ __________ ________ ________ _____ ____ ________ _____.
_______ _________ ____ _______ _______ ____ __________.
__________ ___ ____ _____ _______ __________ ____ ________ _____.
_____ __________ ___ __________ ________ _______ _________ _______ ______ ___ _____ ____.
___ _________ _________ ____ ________ _________ ________ ______ ____ _____ _________ ________.
____ ____ ____ ________ __________ ______ _____ _______ ___ _____.
_____ _________ ______ __________ ___ __________ _______ _____ _______ ________.
_________ ____ ______ ________ _______ ______.
___ ______ _______ ___ __________.
_________ _____ ________ _______ ________ ________ ____ __________ ________.
____ ______ ___ ________ _____.
_______ _______ ____ _____ ________ ______ ______ ___ __________ ___.
________ _________ _____ ____ _________ ______ _________ ________.
________ ___ _______ ___ __________ ____ ______.
_________ _____ __________ __________ ______ ______ _______ _________ _________ ___.
_______ ___ _______ ______ ________ ____ _________ _______ ____ __________ ___.
____ ______ ____ _____ ___ _____ ___ _______ __________ ___ ________.
____ ____ _____ _______ _______ ______ _______.
_________ ___ _______ __________ _______ _______.
_______ __________ __________ ____ ________ _________ ____ ______ ___ _____ ______ ____.
______ ____ _________ ________ _________.
________ _____ _____ ____ ________ ________.
_____ ______ ______ ______ ___ ________ _______ _____ _________ _________ ___ ___.
________ ____ ______ _________ ________ __________ _____ _____ __________ _________.
_____ __________ _______ _______ ___ ________ _____.
Get Full Answer on WhatsApp

Related Questions

If  f'(x_{k})is approximated by:

f'(x_{k}) = af(x_{k+1})+bf'(x_{k+1}),

find the values of a and b. What is the order of approximation?

 

 

Find the solution of the initial boundary value problem , \frac{\partial ^2u}{\partial ^2} = \frac{\partial ^2u}{\partial x^2},0 \leq x \leq1

u(x,0) = sin\pi x,0\leq x\leq 1 \frac{\partial u}{\partial t}(x,0)=0, u(0,t)=u(l,t)=0,t>0

 by using second order explicit method with .h=\frac{1}{4},r=\frac{1}{3} Integrate for one time step. 

Heat conduction equation u_{1} = u_{xx} is approximated by the method: 

u_{m}^{n+1}-u_{m}^{n-1} = frac{2k}{h^2}delta_{k}^{2} u_{m^.}^{n}

Find the order of the method and investigate the stability of this method using Von Neumann method.

Using the recurrence relation

(n+1)P_{n+1}(x)=(2n+1)xP_{n}(x)-nP_{n-1}(x),

prove that

(2n+1)^2\int_{-1}^{1}x^2P_{n}^{2}(x)dx=\frac{2(n+1)^2}{(2n+3)}+\frac{2n^2}{(2n-1)}

Construct Green’s function for the following boundary value problem:

frac{d^2y}{dx^2}+9y = 0 

with y(0) = y(1) = 0.

 

Find the Fourier sine transform of 

f(x)=\left\{\begin{matrix} x, & 0<x<1\\ 2-x,& 1<x<2\\ 0, & x>2 \end{matrix}\right.

Using second order finite differences method with , h = \frac{1}{2}  obtain the system of equations for y_{0^,} y_{1} and y_{2}  for solving the boundary value problem: 

y'' - 5y'+6y = 3

with y(0)-y'(0)=-1 and y(1)+y'(1)=1

IGNOU Solved Assignments Assignments Ignou Projects Project Previous Year Solved Exam Papers Notes Notes Sample Question Papers Sample Question Papers
Login Cart
IGNOU NEWS
Assignment Submission Last Date Extended Till 30 June 2026 Click Here★★★IGNOU June 2026 TEE Date Sheet Released Click Here★★★
Top
📞
Call Support Instant phone assistance State whether the following statements are True or False. Justify your
→
🟢
WhatsApp Chat Fast live messaging
→
Email Us Business enquiries & support
→