MTH401 Midterm Solved Paper Fall 2004 Session 2

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

The differential equation x dy e y dx − is not Bernoulli equation.

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F

Q2. Marks: 2

x f x y x Is homogeneous.

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

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Q4. A set

1, 2 ,..., n y y

Of n linearly dependent solutions, on interval I , of the homogeneous linear nth-order differential equation

x a dx dy x a dx d x a dx d x a n n n n n n L

Is said to be a fundamental set of solutions on the interval I .

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Q5. The differential operator that annihilates

x

x

is D

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Q6.

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

r r dr r r d θ θ θ θ θ = 0

Solution

Q 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)

3 (

1. 1( 1) 1( 2)

r r dr r r d

r r dr r r d

r r dr r r d

r dr r d

r dr d r θ θ θ θ θ

θ θ θ θ θ

θ θ θ

θ θ θ

θ θ θ

question · No: · Question · No: 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 (Just make the equation exact do not solve it further).

x dy x

dx

Solution

It can also be written as 0, x

x xdx x dy

M x y x N x y x

M N x

M N

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

x

dy xdx x dy

N M x g y M x

I F e e

x

x xe dx e x dy

M x y xe N x y e x

M xe N xe

M N

Which shows that equation is exact

Q7.

1. Solve the Bernoulli equation

dy x xy dx +

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

Solution

dy x xy dx dy y

dx x x

put y v

dy dv

dx dx dy dv

dx dx

Then

dv v

dx x x dv v

dx x x

Thus it is linear in “v”.