Show that, for the differential equation

e^mx is a particular integral if m² + am+b = 0. Hence find the value of m so that e^xm is a particular integral of the equation

A mass m , free to move along a line is attracted towards a given point on the line with a force proportional to its distance from the given point. If the mass starts from rest at a distance x₀ from the given point, show that the mass moves in a simple harmonic motion.

Using the method of variation of parameters, solve the equation

Classify the following statements as true or false. Give a short proof of a counter example in support of your answer.

i) The solution of the differential equation    exists, but is not unique.

ii) The differential equation representing all tangents ty = x + t²  at the point (t², 2t) to the parabola y² = 4x is   x(y')² + yy'+1=0 

iii) The p.d.e. au_xx + 2b u_xy + cu,_yy = 0 where a, are constants is irreducible when b² ac = 0 

iv) The functions f₁(x) = cos²x, f₂(x) = sin²x, f₃(x) = sec² x and  f₄(x) = tan²x  are linearly dependent on the interval ] - π/2, π/2 [ 

v) The solution of the second order partial differential equation   involves two arbitrary constants.

Locate and classify the stationary points of the following:

Does the function

  satisfy the requirement of Schwarz’s theorem at (1,1) ? Justify your answer.

Identify the level curves of the following functions:

(i)  

(ii)  

(iii)   x − y

(iv) y / x

Suppose S and C are subsets of   .  S is the unit open sphere with centre at the origin and C is the open cube = 

Which of the following is true. Justify your answer.

(i) S ⊂ C

(ii) C ⊂ S

Find the values of a and b, if

Use double integration of find the volume of the ellipsoid

Evaluate  where C is the curve given by

Find the domain and range of the function f , defined by  Also find two level curves of this function. Give a rough sketch of them.

Check the continuity and differentiability of the function at (0,0) where

Check the local inevitability of the function f defined by f(x,y) =(x²-y²,2xy) at (1,1) Find a domain for the function f in which f is invertible.

Using the Implicit Function Theorem, show that there exists a unique differentiable function g in a neighbourhood of 1 such that g (1) = 2 and F(g( y), y) = 0 in a neighbourhood of (1,2), where

defines the function F. Also find g′( y).

State a necessary condition for the functional dependence of two differentiable functions f and g on an open subset D of . Verify this theorem for the functions f and g, defined by

Find the extreme values of the function

  on the surface 

State Green’s theorem, and apply it to evaluate

Where C is the ellipse 

Find the mass of the solid bounded by z =1 and , z = x² + y² the density function being δ (z,y,x) = | x | .

Find the centre of gravity of a thin sheet with density δ(x, y) = y, bounded by the curves y = 4x² and x = 4.