MTH401 Midterm Past Paper 1 Fall 2004

MIDTERM EXAMINATION · Total Marks: 50 SEMESTER FALL 2004 · MTH401- Differential Equations Duration: 60min · Instructions

2. The Time allowed for this paper is 60 minutes.
3. This examination is closed book, closed notes, closed neighbors; any one found cheating will get zero grades in

the course MTH401 Differential Equations.

4. You are not allowed to use any type of Table for Formulae of Differentiation and integration during your exam.
5. Each objective type question carries 2 marks and each Descriptive question carries 10 marks. So write every

step of the solution of descriptive question to get maximum marks.

Q1. Marks: 2

dy − y

e y

is x

not Bernoulli equation. dx The differential equation

T

F

Marks: 2

Q2. x

fxy

x

Is homogeneous.

T

F

Q3. Population dynamics are not practical application of the first order differential

equations.

T

F

Q4. A set

1, n yy

I n

linearly dependent solutions, on

interval a n x d n y

n dx x a n d n

1 − dx n Of n

th-order differential

equation , of the homogeneous

linear x dy

dx a x y a y L

I

Is said to be a fundamental set of solutions on the

inte l rva T

F

Q5. x

2x D

The differential operator that

ann ilates ih

T

is

F

Q6. r

r d r r r d θ θ θ θ θ

(a)

Define separable form. Just separate the variables of the given differential

equation.

Solution

Q The differential equation of the form d y / d x = f (x, y) is called

d y / d x = h(x)g(y) separable if it can be written in the form

r − θ θ r − dr − r θ θ r − d θ

r − θ θ r − dr r θ θ r − d θ

1) 3 θ (

1. 1( 1( 2)

r −− r dr r − r θ d

r θ d r r θ θ

dr

θ θ

d θ d θ

r

r

Q6. Question No: 6

Marks:10

(b) Check whether the given differential equation is exact or not if not then make it exact also show that it is

exact (

Jus · 2 t m ke t e equation exact do not solve it · urt · a · h · f · her)

x d y x

d x

Solution

It can also be written as xdx x dy

x N , xy 2 − x M xy ,

0,

x M ≠ N x M N x

Thus it is not exact now we apply techniques to make it exact

xdx y − x dy

x N M x

M x gy

dy IF e − ∫

e

− dx e − y − x dy 0 xe

M xy xe − N xy e − x

x N − x − e M e x

M

x N

Which shows that equation is exact

Q7. dy

x

dx xy

(a)

Solve the Bernoulli equation

Just make the given equation linear in v, do not · integrate · Solution

dy

x

dx x y

dy y

dx x x

put y v

dy dv

dx dx

dy dv

y −

dx dx

Then

dv v

dx x x 3

dv v

dx x x

Thus it is linear in “v”.