Mathematical Methods for Physics and Engineering - Matematica.NET

HIGHER-ORDER ORDINARY DIFFERENTIAL EQUATIONSIf the beam is only slightly bent, so that (dy/dx) 2 ≪ 1, where y = y(x) isthedownward displacement of the beam at x, show that the beam profile satisfiesthe approximate equationd 2 ydx = − Wx2 2K .By integrating this equation twice and using physically imposed conditions onyour solution at x =0andx = L/2, show that the downward displacement atthe centre of the beam is WL 3 /(48K).15.4 Solve the differential equationd 2 fdt +6df2 dt +9f = e−t ,subject to the conditions f =0anddf/dt = λ at t =0.Find the equation satisfied by the positions of the turning points of f(t) andhence, by drawing suitable sketch graphs, determine the number of turning pointsthe solution has in the range t>0if(a)λ =1/4, and (b) λ = −1/4.15.5 The function f(t) satisfies the differential equationd 2 fdt +8df2 dt +12f =12e−4t .For the following sets of boundary conditions determine whether it has solutions,and, if so, find them:(a) f(0) = 0, f ′ (0) = 0, f(ln √ 2) = 0;(b) f(0) = 0, f ′ (0) = −2, f(ln √ 2) = 0.15.6 Determine the values of α and β for which the following four functions arelinearly dependent:y 1 (x) =x cosh x +sinhx,y 2 (x) =x sinh x +coshx,y 3 (x) =(x + α)e x ,y 4 (x) =(x + β)e −x .You will find it convenient to work with those linear combinations of the y i (x)that can be written the most compactly.15.7 A solution of the differential equationd 2 ydx +2dy2 dx + y =4e−xtakes the value 1 when x = 0 and the value e −1 when x = 1. What is its valuewhen x =2?15.8 The two functions x(t) andy(t) satisfy the simultaneous equationsdx− 2y = − sin t,dtdy+2x =5cost.dtFind explicit expressions for x(t) andy(t), given that x(0) = 3 and y(0) = 2.Sketch the solution trajectory in the xy-plane for 0 ≤ t