Differential Equations Answers

Given x and y as species whose interaction is governed by

x

0 = x (( 20 0 x + 2y)

y

0 = y (( 50 + x x y),

(i) Identify the type of interaction represented by this system.

(ii) Determine the equilibrium points of this model and state the possible outcomes of

this interaction.

(iii) Linearise the system around each equilibrium point and hence discuss the nature of

the stability of each equilibrium point.

(iv) Sketch the phase portrait of the above system.

Solve, using the method of separation of variables, the PDE

∂u

∂t +

∂u

∂x + 2e

tu = 0

A string of iength L is stretched and fastened to two fix points. Find the solution of

the r.{ave equatiorl (vibrating string) ytt = a^2.yxx, when initial displacernent

y(x,0) = f (x) = b sin (pi.x / t).

also find the Fourier cosine transformation of exp(-x^2)

Evaluate the following functions in differential operator form.

1. D(x²+2x-3) Ans. (2x+2)
2. D²(xe^3x-e^4x) Ans. 9xe^3x+6e^3x-16e^4x

Show that e^x Cosy is Harmonic, find its Conjugate harmonic function.

Find the general solution of the Lagrange's equation 2yzp + zxq = 3xy

A taut string of length 2l , fastened at both ends, is disturbed from its position of equilibrium by imparting to each of its points on an initial velocity of magnitude k(2lx-x^2). Find the displacement function y(x,t)

d^2y/dx^2+4dy/dx-4y=sin3x

Given the initial value problem    x 0 1 − 2x1 − 3x2 = 0 x 0 2 + x1 + x2 = 0 x1(0) = √ 3, x2(0) = 0 , find x2(t). (a) x2(t) = e t 2 sin √ 3 2 t (b) x2(t) = e t  cos 1 2 t + sin 1 2 t  (c) x2(t) = −2e t 2 sin √ 3 2 t (d) x2(t) = 2e t 2 cos √ 3 2 t + sin √ 3 2 t ! (e) x2(t) = 2e t 2 sin √ 3 2 t 

For which values of A , B , C and k is the function yp(x) = x 2 (A x2 + B x + C) a particular solution of the equation 16 y 000 + 9 y 00 = 1458 x k−1 (2x − 1) ? (a) A = 27 , B = 165 , C = −880 , k = 2 (b) A = 27 , B = −219 , C = 1168 , k = 2 (c) A = 27 , B = −219 , C = 1168 , k = 4 (d) A = 54 , B = −411 , C = 2192 , k = 3 (e) A = 54 , B = −411 , C = 2192 , k = 4