MTH401 Midterm Solved Paper Fall 2004 Session 1

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

Order of the differential equation is the highest order derivative in a differential equation.

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Q2. Marks: 2

e− Is integrating factor of the differential equation ( y x y dx y dy

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Q3. If

orthogonal intersect all curves of another family G(x, y, c ) ℑ : H (x, y, c ) ℑ then each curve of the families is said to be orthogonal trajectories of the other.

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Q4. The given functions

f x f x f = x are linearly independent.

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Q5. A set of functions whose wronskian is zero guarantees that set of functions is linearly dependent.

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Q6.

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

4) r r dr r r d θ θ θ θ θ

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)

4)

4)

4. 1( 4) 1( 20)

r r dr r r d

r r dr r r d

r dr r

r dr r d

r dr d r

dr d r θ θ θ θ θ

θ θ θ θ θ

d θ θ θ θ

θ θ θ

θ θ θ

θ θ θ θ

(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).

y x dx xydy

Solution

3 ,

4 , y

y y x dx xydy

M x y y N x y xy

M y N y

M N

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

ln y

dx y x dx xydy

M N y y y

N xy xy

I F e e

y

y xy dx x ydy

M x y xy N x y x y

M xy N xy

M N

Which shows that now equation is exact

Q7.

1. Solve given Bernoulli equation

dy xy y e dx

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

Solution

dy xy y e dx

dy xy y e dx

put y v

dy dv y dx dx

dy dv y dx dx

Then

dv vx e dx

dv vx e dx

Thus it is linear in “v”.