MTH401 Midterm Solved Part 1 with References by Masoom Fairy

MTH401 Mid term past papers (FILE part I) · solved · By masoom fairy · Note: ·  I could not make Neat File due to Much Load shedding. ·  There is an other file because of Large size of this one. · FINALTERM EXAMINATION · SPRING 2007 · MTH401 - DIFFERENTIAL EQUATIONS (Session - 4 )

Q1. The Wronskian of the function (2 marks)

x x W e e is Suppose the functions n f x f x f x possess at least n-1 derivatives on interval I , if

n W f f f is called _________

Linearly dependent

Linearly independent PAGE 110 · MIDTERM EXAMINATION · SEMESTER FALL 2004 · MTH401- Differential Equations

Q1. Marks: 2

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

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Q2. Marks: 2

ex Is integrating factor of the differential equation y · (x · y · 1) dx · +(x · +2 · y)dy

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Q4. Marks:2

f1 ( x )= 1 + x, f2 (x)= x, f2 (x)= x2

The given functions are linearly independent.

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Q5. Marks: 2

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

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(a) Define separable form. Just separate the variables of the given differential equation. (rθ − 4 r +θ −4)dr −( r 2θ + 20r2 −θ −20)dθ = 0

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) (rθ − 4 r +θ −4)dr −( r 2θ + 20r2 −θ −20)dθ = 0

(rθ − 4 r +θ −4)dr = ( r 2θ + 20r2 −θ −20)dθ

[ r (θ − 4) +1(θ − 4)]dr = r 2 (θ + 20)−1(θ + 20) dθ

( r + 1)(θ − 4)dr = ( r 2 −1)(θ + 20)dθ ( r + 1) (θ + 20) ( r 2 − 1) dr (θ −4) dθ 1 dr = ( (θ +

20. ) dθ r − 1 θ −4

(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).