MTH401 Midterm Solved Paper Fall 2004 Session 3

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 to be ordinary differential equation if it contains only ordinary derivatives with

respect to single variable.

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Q2. Marks: 2

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

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Q3. Logistic equations are applications of non-linear equations.

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Q4. The given functions

5, cos sin f f x f x are linearly independent.

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Q5. A set of functions whose wronskian is zero then set of functions may or may not be dependent.

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Q6.

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

sec sin sin dy y y dx + y

Solution

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

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

sec sin sin

sec sin sin

sec sin sin

sec sin cos cos sin sin cos cos sin

sec 2cos sin

cos 2cos sin

cos sin 2

cos cos dy y y y dx

dy y y y dx

dy y y y dx

dy y y y y dx

dy y y dx

dy xdx y y

dy xdx y

ec ydy xdx y

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

exact (Just make the equation exact do not solve it further).

cot 2 cos e dx e y y ecy dy

Solution

cot 2 cos

cot 2 cos

0, cot y

y e dx e y y ecy dy

M x y e N x y e y y ec

M N e y

M N y

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

cot lnsin cot 2 cos

cot cot

si y

ydy y e dx e y y ecy dy

N M e y y M e

I F e e y n

sin cos

sin , cos

cos , cos y

y e ydx e y y dy

M x y e y N x y e y y

M e y N e y

M N

Which shows that equation is exact

Q7.

1. Solve Bernoulli equation

dy y y dx − 1)

(Just make the given equation linear in v, do not integrate)

Solution

1)

1) dy xy y dx

dy xy y dx

put y v

dy dv y dx dx

dy dv y dx dx

1)

2( Then

dv vx dx

dv vx dx −1)

(b) The radioactive isotope of the lead, Pb-209, decay at a rate proportional to the amount present at any time and

has a half-life of 4 hours. If 2 grams of the lead is present initially, how long will it take for 80% of the lead to

decay? (Just make the model of the radioactive decay as well as describe the given conditions do not solve