MST224 Mathematical methods - Open University

1 MST224 Mathematical methodsSecond-order differential equationsThis publication forms part of an Open University module. Details of this and otherOpen University modules can be obtained from the Student Registration and Enquiry Service, TheOpen University , PO Box 197, Milton Keynes MK7 6BJ, United Kingdom (tel. +44 (0)845 300 6090;email you may visit the Open University website where you can learnmore about the wide range of modules and packs offered at all levels by The Open purchase a selection of Open University materials visit , or contact OpenUniversity Worldwide, Walton Hall, Milton Keynes MK7 6AA, United Kingdom for a brochure(tel. +44 (0)1908 858779; fax +44 (0)1908 858787; email to readerMathematical/statistical content at the Open University is usually provided to students inprinted books, with PDFs of the same online. This format ensures that Mathematical notationis presented accurately and clearly.))

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4 Ltd, Amersham, 978 1 7800 7479 preliminary Requirement for two arbitrary Linearity and superposition82 Homogeneous differential Two simple Solution in the general General procedure and further Damped harmonic oscillators223 Inhomogeneous differential General method of The method of undetermined Exceptional Combined cases404 Initial conditions and boundary Initial-value Boundary-value problems445 Resonance48 Solutions to exercises55 Acknowledgements66 Index67 IntroductionIntroductionIn this unit we move from first-orderdifferential equations to second-orderTheorderof a differentialequation is defined in Unit second-order differentialequation may or may notinclude a first equations , that is, differential equations involving a second (butno higher) derivative. Examples of such equations ared2ydx2 3dydx+ 2y= 4exand 3d2ydx2+y= differential equations play a central role in the physicalsciences.

5 They are found, for example, in laws describing mechanicalsystems, wave motion, electric currents and quantum take a simple case, consider a particle of massmthat moves in onedimension along thex-axis. At any given timet, the particle s position isx(t), and its velocity and acceleration are given by the derivativesdx/dtandd2x/dt2. There are no general laws for the position or velocity of theparticle, but there is a very important law for its acceleration:Newton sDon t worry if you have not metNewton s second law before: theessential aims of this unit do notrely on lawtells us thatmass acceleration = force,which implies thatmd2xdt2=F,(1)whereFis the force acting on the particle. The force need not beconstant, and may vary with the positionxor the velocitydx/dtof theparticle. So, depending on the precise details, we get a second-orderdifferential equation forxas a function oft, and the solution of thisequation tells us how the particle can system known as asimple harmonic oscillatorprovides a goodexample.

6 Here, a particle of massmis suspended at the lower end of aspring that is attached to a fixed support (Figure 1). The particle movesxx= 0mFFigure 1A particle ofmassmmoves along thex-axis subject to a forceFprovided by a spring andgravityup and down along a verticalx-axis, subject to a forceFprovided by thespring and gravity. If the system is left to settle, the particle comes to restat a point of equilibrium, which we labelx= 0. Because the particle doesnot spontaneously move away from this position, we can inferthatF= 0whenx= the particle is displaced fromx= 0, the forceFtends to draw theparticle back towardsx= 0. We consider the case where the force isproportional to the displacement from equilibrium, and takeF= kx,(2)wherekis a positive constant. The negative sign in this equation ensuresthat the force always acts in a direction that tends to restore the particleto its equilibrium equations (1) and (2) together, we get the differential equationmd2xdt2= differential equationsRecalling thatk>0 andm >0, we can also express this asd2xdt2= 2x,(3)where = k/mis a positive constant.

7 Equation (3) is called theequation of motionof a simple harmonic oscillator. It is a second-orderdifferential equation whose solution tells us how the particle can unit develops systematic techniques to solve equations like this. Forthe moment, we will simply guess the solution and check that it know thatddt(sint) = costandddt(cost) = sint,sod2dt2(sint) = sintandd2dt2(cost) = other words, taking the second derivative of a sine or cosine functiongives the same function back again, but with a minus sign. This is veryclose to the behaviour needed to solve equation (3). We therefore try afunction of the formx(t) =Csin( t) +Dcos( t),(4)whereCandDare any constants, and is the constant in equation (3).Differentiating this function once, and then again, we getdxdt=C cos( t) D sin( t),d2xdt2= C 2sin( t) D 2cos( t)= function in equation (4) does indeed satisfy equation (3). In fact, itis the general solution of this differential that our solution involves two constants,CandD, whose values arenot specified.

8 These constants have arbitrary values, and they are calledarbitrary constants. It is typical for the general solutions of a second-orderdifferential equation to havetwoarbitrary constants. The values of theseconstants depend on how the system is released, and you will see how theyare determined later in this unit. To take a definite case, suppose thatC= 2,D= 3 and = 4, when measured in suitable units. Then we havethe particular solutionx(t) = 2 sin(4t) + 3 cos(4t),(5)and this is plotted in Figure 2. The wiggles in this graph correspond to theoscillations performed by a system like the particle on the end of thespring in Figure 4 2 2A graph of equation (5), which is a particular solution ofequation (3)The simple harmonic motion described by equation (4) continues forever,but we know from everyday experience that oscillations generally die awayafter a while. We can get a more realistic description by including anadditional force in our model one that will dampen the oscillations take the additional force to be dx/dt, where is a positiveconstant, leading to the differential equationmd2xdt2= kx dxdt.

9 (6)This is the equation of motion of adamped harmonic oscillator. Later inthis unit you will see that (in appropriate circumstances) it has solutionsthat oscillate but diminish and eventually die can also add in another force,f(t), which is applied to the particle bysome external agency. We assume that this force is known directly as afunction of time (and does not depend on the position or velocity of theparticle.) Under these circumstances, Newton s second lawleads to thedifferential equationmd2xdt2= kx dxdt+f(t),(7)and this is the equation of motion of aforced damped harmonic the external force is oscillatory, the response of the system may dependvery sensitively on the frequency of the external force. This phenomenon iscalledresonance; it will be explored at the end of this oscillators are everywhereHarmonic oscillators play a central role in physics and itsapplications. If a system performs small oscillations about anequilibrium point, then it is generally a good approximation to modelit as a harmonic oscillator, including the additional termsinequations (6) and (7) when differential equationsIt should come as no surprise that the to-and-fro motion of apendulum clock can be modelled by a harmonic oscillator.

10 On asmaller scale, vibrating molecules and vibrating crystals are alsomodelled as harmonic similar to (7) are also used to describe the oscillations ofcurrents in electrical circuits that allow radios to be tuned to selectedstations. Moreover, each frequency in an electromagnetic field can beregarded as a harmonic oscillator, and this is a key insight used inadvanced physics when electromagnetic fields are treated more everyday example is provided by the suspension system of amountain bike (or any other vehicle). The rider is protected from thevibrations caused by a rough track by a rugged suspension system,and this can also be modelled by equation (7).Study guideThis unit requires no previous knowledge beyond that needed forUnit 2,apart from some familiarity with complex numbers. The relevant materialon complex numbers was reviewed in Unit differential equations discussed in this unit all belong to one broadclass: they are alllinear constant-coefficient second-order differentialequations.