2. Higher-order Linear ODE’s

1 2. Higher-order Linear ODE s2A. Second- order Linear ODE s: General the right below is an abbreviated form of the ODE on the left:(*)y +p(x)y +q(x)y=r(x)Ly=r(x) ;whereLis thedifferential operator:L=D2+p(x)D+q(x).a) Ifu1andu2are any two twice-differentiable functions, andcis a constant, thenL(u1+u2) =L(u1) +L(u2)andL(c u) =c L(u).Operators which have these two properties are calledlinear. Verify thatLis Linear , ,that the two equations are ) Show that ifypis a solution to (*), then all other solutions to (*) can be written inthe formy=yc+yp,whereycis a solution to theassociated homogeneous equationLy= ) By eliminating the constants, find a second- order Linear homogeneous ODE whosegeneral solution isy=c1ex+ ) Verify for this ODE that the IVP consisting of the ODE together with the initialconditionsy(x0) =y0,y (x0) =y 0y0, y 0constantsis always ) By eliminating the constants, find a second- order Linear homogeneous ODE whosegeneral solution isy=c1x+ )

2 Show that there is no solution to the ODE you found in part (a) which satisfies theinitial conditionsy(0) = 1, y (0) = ) Why doesn t part (b) contradict the existence theorem forsolutions to second-orderlinear homogeneous ODE s? (Book: Theorem 2, p. 110.) the ODEy +p(x)y +q(x)y= ) Show that ifpandqare continuous for allx, a solution whose graph is tangent tothex-axis at some point must be identically zero, , zero for ) Find an equation of the above form havingx2as a solution, by calculating itsderivatives and finding a Linear equation connecting them. Why isn t part (a) contradicted,since the functionx2has a graph tangent to thexaxis at 0?

3 That the following pairs of functions are linearly independent, by calculatingtheir )em1x, em2x, m16=m2b)emx, xemx(canm= 0?) |x|. (Sketch the graph ofy2.)a) Show thatW(y1, y2) 0 ( , is identically zero).b) Show thaty1andy2are not linearly dependent on any interval (a, b) containing doesn t this contradict theorem 3b, p. 116 in your book? two solutions ofy +p(x)y +q(x)y= ) Prove thatdWdx= p(x)W, whereW=W(y1, y2), the ) Prove that ifp(x) = 0, thenW(y1, y2) is always a ) Verify (b) by direct calculation fory +k2y= 0, k6= 0, whose general solution isy1=c1sinkx+ Reduction of a second solutiony2toy 2y +y= 0, given that one solution isy1=ex, bythree methods:a) puttingy2=uexand determiningu(x) by substituting into the ODE;b) determiningW(y1, y2) using Exercise 2A-7a, and from this gettingy2;c) by using the general formulay2=y1 1y21e p dxdx.

4 D) If you don t get the same answer in each case, account for the differences. (What isthe most general form fory2?) Exercise 2B-1, prove that the general formula in part (c) for a second solution givesa functiony2such thaty1andy2are linearly independent. (Calculate their Wronskian.) the method of reduction of order (as in 2B-1a) to find a second solution tox2y + 2xy 2y= 0,given that one solution isy1= the general solution on the interval ( 1,1) to the ODE(1 x2)y 2xy + 2y= 0,given thaty1=xis a 2. Higher-order ODE S32C. Second- order Linear ODE s with Constant the general solution, or the solution satisfying the given initial conditions, toeach of the following:a)y 3y + 2y= 0b)y + 2y 3y= 0;y(0) = 1, y (0) = 1c)y + 2y + 2y= 0d)y 2y + 5y= 0;y(0) = 1, y (0) = 1e)y 4y + 4y= 0;y(0) = 1, y (0) = by using the Wronskian criterion thateaxcosbxandeaxsinbxare linearlyindependent.

5 Are there any restrictions on the constantsaandb? +cy + 4y= 0, cconstant. For each statement below, tell for whatvalue(s) ofcit holds (indicate reasoning):a) the equation has oscillatory solutionsb) all solutions are damped s equidimensional equationis the ODEx2y +pxy +qy= 0;p, ) Show that settingx=etchanges it into an equation with constant ) Use this to find the general solution tox2y +xy +y= equationmx +cx +kx= 0 represents the motion of a damped spring-masssystem. (The independent variable is the timet.)How are the constantsm, c, krelated if the system is critically damped ( , just on theedge of being oscillatory)?

6 That the angle of the pendulum swinging with small amplitude (so you canuse the approximation sin ) approximately obeys a second- order ODE with constantcoefficients. UseL= length,m= mass,damping =mcd dt,for some constantcIf the motion is undamped, ,c= 0, express the period in terms ofL, m, and thegravitational each of the following, tell what you would use as the trialsolution in determininga particular solution by the method of undetermined coefficientsa)y + 2y + 2y=x+exb)y 4y = cos 2xc)y + 4y= 3 cos 2xd)y 2y +y= 3exe)y 3y + 2y=e x+ 3e2xf)y 6y + 9y= the general solution, or the solution satisfying the given initial conditions.

7 A)y 6y + 5y=exb)y + 4y= 2 cosx, y(0) = 0, y (0) = 1c)y +y +y= 2x exd)y y=x2, y(0) = 0, y (0) = the ODEy +p(x)y +q(x)y=r(x).a) Prove that ifyiis a particular solution whenr=ri(x), (i= 1,2), theny1+y2is aparticular solution whenr=r1+r2. (Use the ideas of Exercise 2A-1.)b) Use part (a) to find a particular solution toy + 2y + 2y= 2x+ series RLC-circuit is modeled by either of the ODE s (the second equation isjust the derivative of the first)Lq +Rq +qC=E,Li +Ri +iC=E ,whereq(t) is the charge on the capacitor, andi(t) is the current in the circuit;E(t) is theapplied electromotive force (from a battery or generator),and the constantsL, R, Carerespectively the inductance of the coil, the resistance, and the capacitance, measured insome compatible system of ) Show that ifR= 0 andE= 0, thenq(t) varies periodically, and find the period.

8 (AssumeL6= 0.)b) AssumeE= 0; how mustR, L, Cbe related if the current oscillates?c) IfR= 0 andE=E0sin t, then for a certain 0, the current will have large amplitudewhenever 0. What is the value of 0. (Indicate reason.)2D. Variation of a particular solution by variation of parameters:a)y +y= tanxb)y + 2y 3y=e xc)y + 4y= s equation of order pisx2y +xy + (x2 p2)y= , two independent solutions forx >0 arey1=sinx xandy2=cosx x, x > the general solution tox2y +xy + (x2 14)y=x3/2cosx . the ODEy +p(x)y +q(x)y=r(x).a) Show that the particular solution obtained by variation of parameters can be writtenas the definite integraly= xa y1(t)y2(t)y1(x)y2(x) W(y1(t), y2(t))r(t)dt.

9 (Write the functionsv1andv2(in the Variation of Parameters formula) as definite integrals.)b) If instead the particular solution is written as an indefinite integral, there are arbitraryconstants of integration, so the particular solution is notprecisely defined. Explain why thisdoesn t use variation of parameters to find a particular solution, ratherthan the method of undetermined coefficients?SECTION 2. Higher-order ODE S52E. Complex NumbersAll references are to Notes C: Complex to polar form: a) 1 +ib) 3 i1 +iin the forma+biby two methods: one using the Cartesian formthroughout, and one changing numerator and denominator to polar form.

10 Show the twoanswers * Show the distance between any two complex pointsz1andz2is given by|z2 z1|. two laws of complex conjugation:for any complex numberszandw, a)z+w=z+wb)zw=zw .2E-5.* Supposef(x) is a polynomial withrealcoefficients. Using the results of 2E-4, showthat ifa+ibis a zero, then the complex conjugatea ibis also a zero. (Thus, complexroots of a real polynomial occur in conjugate pairs.)2E-6.* Prove the formulaei ei =ei( + )by using the definition ( euler s formula (9)),and the trigonometric addition each of the following two ways: by changing to polar form, and also byusing the binomial ) (1 i)4b) (1 +i 3)32E-8.