MTH401 Past Papers Combined

MTH401 · Final Term Examination – Spring 2006 · Time Allowed: 150 Minutes

Q1. Marks : 5

Determine the singular points of the differential equation and classify them as regular or irregular.

( x 2 − 9) 2 y′′ + ( x + 3) y′ + 2 y =

Q2. Marks : 10

Solve the given Bernoulli equation

dy + y = y dx

Q3. Marks : 2

Which of the following is singular point of the equation

2 x 2 y′′ + 10 xy′ + ( x − 1) y = 0

x = 0 x = 2 x = 1

Q4. Marks : 10

Find the eigen value and eigen vector of the following system of linear differential equation

⎛ 9 −12 ⎞ X ′ = ⎜ ⎟ X ⎝ 3 −3 ⎠ x

p

p

Q5. Marks : 10

Find solution of the differential equation

9 y ′ + y =

in the form of a powers series in x .

Q6. Marks : 2

The non-trivial solution of the system exists only when

det( A − λ I ) = 0

• True
• False

Q7. Marks : 2

The form of the particular solution for the differential equation y′ − y = x 2e 3 x is

y p =

Ae + B1 x + B0

y = ( Ax + B)ex

y = ( Ax2 + Bx + C )e3 x

• None of above

Q8. Marks : 10

Solve the following homogeneous system of differential equations dx − 7 x + dy = 3et dt dt 3 dx − 2x + dy = 2et dt dt

(Just find the general solution of the equation). y′′ − 2 y′ + y = 0

Q10. Marks : 2

Which of the following is Legendre’s Equation

d 2 y dy (1 −

x ) dx − x + 7 y = 0 dx

2 d y dy (1 −

x ) dx 2 − 7 x + 35 y = 0 dx

= d ⎡(1 − x 2 ) =dy ⎤ + 6 y = 0 dx dx

• All of three equations are legendre's equation

Q11. Marks : 2

The differential equation (2x 2 y − 2x 3 )dy + (4 x 3 − 6x 2 y + 2 xy 2 )dx = 0 is

• Separable
• Exact
• Linear
• Bernoulli’s