MTH401 Midterm Past Paper 2 Fall 2004

MIDTERM EXAMINATION · SEMESTER FALL 2004 · MTH401- Differential Equations · 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

A differential equation said respect to single variable. to be ordinary differential equation if it contains only ordinary derivatives

with

T

F

Q2. Marks: 2

A solution of the differential equation of the form y= f (x) is calle d the implicit solution.

T

F

Q3. L

ogistic equations are applications of non-linear

equations.

T

F

Q4. cos

sin 5, f f 1 f x

The given functions are linearly independent.

T

F

Q5. A

set of functions whose wronskian is zero then set of functions may or may not be

dependent.

T

F

Q6. y

(a)

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

equation. sec st d

sin sin y

y

dx − y

Solution

separable if it can be written in the form The differential equation of the form d y / d x = f (x, y) is called

d y / d x = h(x)g(y)

sin sin dy

y

dx x − y x y sec

sin sin dy

y

dx x y − x − y sec

sin sin dy

y

dx x y − x − y sec

sin cos cos sin sin cos cos dy

y

dx y y − y sec sin y

2cos sin dy

y

dx y sec

cos

sin dy

y y xdx

2cos

dy

cos xdx

sin 2 y

cos ec ydy xdx cos

Just make the equation exact do not solve it · urther) · f · Solution

cot 2 cos e dx e y y ecy dy (b) Check whether the given differential equation is exact or not if not then make it exact also show that it is

exact edx e cot y 2 y cos ecy dy 0

M x , y e , N x , y e cot y 2 y cos e y c

y M 0, N e cot y

y M ≠ N

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

cot 2 cos e dx e y y ecy dy 0

N M x e cot y

y cot y

x e M

cot

lnsin s i ydy

y IF e e n y

cos d x e y y d y e sin y

s i n c o s M xy e y N xy e y 2 y

cos y N e y x e cos M y

M N

y

Which shows that equation is exact

Q7. dy

y

dx y 1) x −

(a) Solve Bernoulli equation

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

dy

xy

dx y x − 3 ( 1 )

dy

xy y

dx − 2 − 1 )

put y v

dy dv

y

dx dx − 2

Then dy dv

y

dx dx

1 dv

vx x − )

2 dx

dv

1 ) 2 ( vx x −

dx

The r as a hala f dioactive isotope of the lead, Pb-209, decay at a rate proportional to the amount present at any tim -life of 4 hours. If 2 grams of the lead is present initially, how long will it take for 80% of the lead to h

(b)

deca y ?

(J

further) ust make the model of the radioactive decay as well as describe the given conditions do not

solve

Solution

e and