{F} Obtain two linearly independence solution valid near the origin for the following equation
π₯π¦"+ 3π₯π¦β² + ( 1+4π₯2) π¦=0
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
βu
βt +
βu
βx + 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.
Solve using the method of separation of variables the pde partial du/dt+du/dx+2e^tu=0
Show that e^x Cosy is Harmonic, find its Conjugate harmonic function.
Find the general solution of the Lagrange's equation 2yzp + zxq = 3xy
(d^2-4)y=4x-3e^x
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)