Mathematical Methods for Physics and Engineering - Matematica.NET

15.4 EXERCISES15.9 Find the general solutions of(a)(b)d 3 y dy− 12 +16y =32x − 8,dx3 ( dx )(d 1 dy1+(2a coth 2ax)dx y dxydydx)=2a 2 ,where a is a constant.15.10 Use the method of Laplace transforms to solve(a)d 2 fdt +5df2 dt +6f =0, f(0) = 1, f′ (0) = −4,(b)d 2 fdt +2df2 dt +5f =0, f(0) = 1, f′ (0) = 0.15.11 The quantities x(t), y(t) satisfy the simultaneous equationsẍ +2nẋ + n 2 x =0,ÿ +2nẏ + n 2 y = µẋ,where x(0) = y(0) = ẏ(0) = 0 and ẋ(0) = λ. Show thaty(t) = ( 1 2 µλt2 1 − 1 nt) exp(−nt).315.12 Use Laplace transforms to solve, for t ≥ 0, the differential equationsẍ +2x + y =cost,ÿ +2x +3y =2cost,which describe a coupled system that starts from rest at the equilibrium position.Show that the subsequent motion takes place along a straight line in the xy-plane.Verify that the frequency at which the system is driven is equal to one of theresonance frequencies of the system; explain why there is no resonant behaviourin the solution you have obtained.15.13 Two unstable isotopes A and B and a stable isotope C have the following decayrates per atom present: A → B, 3s −1 ; A → C, 1s −1 ; B → C, 2s −1 . Initially aquantity x 0 of A is present, but there are no atoms of the other two types. UsingLaplace transforms, find the amount of C present at a later time t.15.14 For a lightly damped (γ < ω 0 ) harmonic oscillator driven at its undampedresonance frequency ω 0 , the displacement x(t) attimet satisfies the equationd 2 x dx+2γdt 2 dt + ω2 0x = F sin ω 0 t.Use Laplace transforms to find the displacement at a general time if the oscillatorstarts from rest at its equilibrium position.(a) Show that ultimately the oscillation has amplitude F/(2ω 0 γ), with a phaselag of π/2 relative to the driving force per unit mass F.(b) By differentiating the original equation, conclude that if x(t) is expanded asa power series in t for small t, then the first non-vanishing term is Fω 0 t 3 /6.Confirm this conclusion by expanding your explicit solution.15.15 The ‘golden mean’, which is said to describe the most aesthetically pleasingproportions for the sides of a rectangle (e.g. the ideal picture frame), is givenby the limiting value of the ratio of successive terms of the Fibonacci series u n ,which is generated byu n+2 = u n+1 + u n ,with u 0 =0andu 1 = 1. Find an expression for the general term of the series and525