D^2x + 3x - 2y = 0, D^2x + D^2y - 3x + 5y = 0}, where D = d/dt If x = 0, y = 0; Dx = 0, Dy = 2 when t = 0, find x and y when t = 1/2.

Question

To small Oscillations of a certain system with two degrees of freedom are given by the equations

D²x + 3x - 2y = 0, D²x + D²y - 3x + 5y = 0}, where D = d/dt

If x = 0, y = 0; Dx = 0, Dy = 2 when t = 0, find x and y when t = 1/2.

Answer 1

Re–write the given set of simultaneous equations as 

 (D² + 3)x – 2y = 0 …(1)

(D² – 3)x + (D² + 5)y = 0 …(2) 

In order to make above equations separately in x(t) and y(t) i.e. to eliminate x, first operate these equations by (D² – 3) and (D² + 3) respectively and then subtract (1) from (2), we get 

 [(D² + 3)(D² + 5) + 2(D² – 3)]y = 0 

On simplification, 

(D⁴ + 10D² + 9)y = 0 … (3) 

Corresponding auxiliary equation becomes 

D⁴ + 10D² + 9 = 0 or (D² + 1)(D² + 9) = 0 i.e. D = ± i, ± 3i 

Thus, y(C.F.) = (a₁cos t + a₂sint) + (a₃cos3t + a₄sin3t) … (4) 

To find x, eliminate y from (1) and (2). Operate (1) by (D² + 5) and multiply (2) by 2 and then add the two, 

(D⁴ + 10D² + 9)x = 0 (an equation identical to (3)) i.e. D = ± i, ± 3i ...(5) 

Thus, x(C.F.) = (b₁cost + b₂sint) + (b₃cos 3t + b₄sin3t) …(6) 

To find the relation between constants involved in (4) and (6)

Substitute values of x and y in either of the given equations, say in (1), we get 2(b₁ – a₁)cost + 2(b₂ – a₂)sint – 2(3b₃ + a₃)cos3t – 2(3b₄ + a₄)sin3t = 0 …(7) 

Which must holds for all t. 

On equating co–efficient of cost, sint, cos3t, sin3t, we get