MTH401 Final Term Solved Subjective

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

y y

about · such that · and

n n a x

n y x

Solution:

is a function of y

Case2:

Then find the integrating factor in both cases.

Solution:

u xM yN

Q6. Find eigen vector of

A=

,corresponding eigen value

( 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

0 ..........(1) y P x y Q x y

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

t

t dx dy x e dt dt

dx dy x e dt dt

lec 36 example 1

in decoupled form.

Determinants are

Therefore, in decpoupled form, we get t

t

t

t

t t

t t

t

t

t

t dx dy x e dt dt

dx dy x e dt dt

D x Dy e

D x Dy e

D D e D D e

D D e D D e

D D e D x D D e D

D D D e y D D D e

Q10. When Frobenius’ Theorem is used in Differential Equation

a x y a x y a x y