MTH401 Final Term Past Papers 2011 by Shining Star

Rana Abubakar Khan

If you want to learn computer programming then contact with me

JOIN FB GROUP FOR VU HELP · MTH401-Diffeerential Equations · Fall 2011 FINAL Paper 23 FEB, 2011 · BY · *~*SHINING STAR*~*($$)

Q1. Write down the system of differential equations

in form of

Solution:

t X X t 6 , dx dy x y t x y t dt dt

X AX F t

Q2. Solve the system of differential equations

by systematic elimination.

Solution:

Operate (ii) by , we get

1. ..........(

Add (i) and (iii), we get

Auxiliary equation is dy x Dy x i dt

dx y y Dx ii dt

D

Dy D x iii

Dy x

Dy D x

D x x

D x

m

m

x t

Put this in (i), we get

Integrate both sides, we get t t

t t

t t

t t c e c e

Dy c e c e

Dy c e c e

y t c e c e

Q3. Find a series solution for the differential equation

about · such that · and

dy dx x y dt dt

y y

n n a x

n y x ( 2)

( 2) 0

Add two times row 1 in row 2

Choosing k 1, we get k

therefore, eigen vector is k k

k k

V

Q8. Under which conditions linear independence of the solutions for the differential equation

is guaranteed?

Solution:

Linear independence is guaranteed in case when the Wronskian of the two solutions is not equal to zero.

Q9. Write the homogenous system of differential equations

lec 36 example 1

in decoupled form.

1. .........(1)

y P x y Q x y

t

t dx dy x e dt dt

dx dy x e dt dt