The Wronskian - math.usm.edu

1 Jim LambersMAT 285 Spring Semester 2012-13 Lecture 16 NotesThese notes correspond to Section in the WronskianNow that we know how to solve a linear second - order homogeneous ODEy +p(t)y +q(t)y= 0in certain cases, we establish some theory about general equations of this form. First, we introducesome notation. We define asecond- order linear differential operatorLbyL[y] =y +p(t)y +q(t) , a initial value problem with a second - order homogeneous linear ODE can be stated asL[y] = 0, y(t0) =y0, y (t0) = state a result concerning existence and uniqueness of solutions to such ODE, analogous tothe Existence-Uniqueness Theorem for first- order (Existence-Uniqueness)The initial value problemL[y] =y +p(t)y +q(t)y=g(t), y(t0) =y0, y (t0) =z0has a unique solution on an open intervalIcontaining the pointt0ifp,qandgare continuous onI.

2 The solution is twice differentiable , suppose that we have obtained two solutionsy1andy2of the equationL[y] = 0. Theny 1+p(t)y 1+q(t)y1= 0, y 2+p(t)y 2+q(t)y2= +c2y2, wherec1andc2are constants. ThenL[y] =L[c1y1+c2y2]= (c1y1+c2y2) +p(t)(c1y1+c2y2) +q(t)(c1y1+c2y2)=c1y 1+c2y 2+c1p(t)y 1+c2p(t)y 2+c1q(t)y1+c2q(t)y2=c1(y 1+p(t)y 1+q(t)y1) +c2(y 2+p(t)y 2+q(t)y2)= have just established the following (Superposition)Lety1andy2be solutions of the equationL[y] = 0. Then, for anyconstantsc1andc2, the linear combinationc1y1+c2y2is also a that we can obtain infinitely many solutions ofL[y] = 0 from two solutionsy1andy2, itis natural to ask whetherallsolutions ofL[y] = 0 are of the formc1y1+c2y2, for constantsc1andc2.

3 For this to be the case, it is necessary to be able to satisfy any given initial (t) =c1y1(t) +c2y2(t). Substituting this solution into the initial conditionsy(t0) =y0andy (t0) =z0, we obtain the system of equationsc1y1(t0) +c2y2(t0) =y0,c2y 1(t0) +c2y 2(t0) =z0,or, in matrix vector form,Y(y1,y2)(t0)c=u0,whereY(y1,y2)(t0) =[y1(t0)y2(t0)y 1(t0)y 2(t0)],c=[c1c2],u0=[y0z0].From linear algebra, this system has a unique solution for any right-hand sideu0if and only ifthe coefficient matrixY(y1,y2)(t0) has a nonzero determinant. That is, we must haveW(y1,y2)(t0) = detY(y1,y2)(t0) =y1(t0)y 2(t0) y2(t0)y 1(t0)6= functionW(y1,y2)(t), which is a function oftbut depends on the solutionsy1(t) andy2(t),is called theWronskianofy1andy2.

4 If the Wronskian is nonzero, then we can satisfy any initialconditions. We have just established the following two solutions ofL[y] = 0. Then there exist constantsc1andc2so thaty(t) =c1y1(t) +c2y2(t)satisfiesL[y] = 0 and the initial conditionsy(t0) =y0, y (t0) =z0if and only if the WronskianW=y1y 2 y2y 1is nonzero can actually make a stronger statement: if the Wronskian is nonzero, then not only can weobtain a solution for any initial conditions, but we can actually describeallsolutions of the initialvalue problem. That is, there are no other solutions that are not a linear combination ofy1andy2. This is formally stated in the following solutions ofL[y] = 0. Then every solution ofL[y] = 0 is of the formy(t) =c1y1(t) +c2y2(t)if and only if the Wronskian ofy1andy2is nonzero at a the linear combinationy(t) =c1y1(t) +c2y2(t)describesallsolutions of the equationL[y] = 0, it is called thegeneral solutionof this also say that the solutionsy1andy2form afundamental set of soultionsof the the ODEy + 4y + 4y= solutions of this ODE arey1(t) =e 2tandy2(t) =te 2t.

5 Their Wronskian isW(y1,y2)(t) =e 2t(te 2t) te 2t(e 2t) =e 2t(e 2t 2te 2t) te 2t( 2e 2t)=e 4t 2te 4t+ 2te 4t=e 4t,which is nonzero. Therefore,y1andy2form a fundamental set of solutions, and all solutions of theequation are of the formc1y1+ , when solving the constant-coefficient equationy +py +qy= 0,where the roots 1and 2of the characteristic equation 2+p +q= 0 are real and distinct, wecalled the solutiony(t) =c1e 1t+c2e 2tthe general solution. This is justified because at any timet,W(e 1t,e 2t)(t) =e 1t(e 2t) e 2t(e 1t) = 2e 1te 2t 1e 1te 2t=e 1te 2t( 2 1)6= 0,so by the preceding theorems,y(t) actually is the general solution in the sense in which we havejust defined it, and{e 1t,e 2t}is a fundamental set of solutions.

6 If 1= 2, however, we donothave a fundamental set of solutions, as the Wronskian would be zero. Later, we will learn how toobtain a second solution which, paired withe 1t, will form a fundamental set of the more general linear homogeneous second - order ODE, we can obtain a fundamental setof solutions by solving two specific initial value (t) andq(t) be continuous on an open intervalIcontaining a pointt0. Lety1bethe unique solution of the ODEL[y] =y +p(t)y +q(t)y= 0with initial conditionsy(t0) = 1, y (t0) = 0,and lety2be the unique solution ofL[y] = 0 with initial conditionsy(t0) = 0, y (t0) = a fundamental set of solutions ofL[y] = prove this theorem, we simply note thatW(y1,y2)(t0) = det([y1(t0)y2(t0)y 1(t0)y 2(t0)])= det([1 00 1])= 1(1) 0(0) = 16= 0,3which proves that{y1(t),y2(t)}is a fundamental set of the ODEy 3y + 2y= characteristic equation is 2 3 + 2 = 0, which has roots 1= 1 and 2= 2.

7 Therefore, thegeneral solution isy(t) =c1et+ (0) =c1+c2, y (0) =c1+ 2c2,we find that the solutionw1(t) = 2et e2tsatisfies the initial conditionsw1(0) = 1, w 1(0) = , the solutionw2(t) = et+e2tsatisfies the initial conditionsw2(0) = 0, w 2(0) = {w1,w2}is a fundamental set of solutions of the conclude by deriving a simple formula for the Wronskian of any fundamental set of solutions{y1,y2}ofL[y] = 0. Because they are solutions, we havey 1+p(t)y 1+q(t)y1= 0,y 2+p(t)y 2+q(t)y2= the first equation byy2and the second equation byy1, and then subtracting the firstequation from the second , we obtainy 2y1 y 1y2+p(t)(y1y 2 y2y 1) = noting thatddt[W(y1,y2)(t)] =ddt[y1y 2 y2y 1] =y1y 2+y 1y 2 y 2y 1 y2y 1=y 2y1 y 1y2,we obtainddt[W(y1,y2)(t)] +p(t)W(y1,y2)(t) = is a first- order separable linear equation, which has the solutionW(y1,y2)(t) =cexp[ p(t)dt],wherecis an arbitrary constant.

8 This is summarized in the following (Abel s Theorem)Letp(t) andq(t) be continuous on an open intervalI, and lety1andy2be solutions of the ODEL[y] =y +p(t)y +q(t)y= the WronskianW(y1,y2)(t) is given byW(y1,y2)(t) =cexp[ p(t)dt],wherecis a constant that depends ony1andy2. Furthermore,W(y1,y2)(t) is either never zero inI(ifcis nonzero) or is zero for allt I(ifc= 0).It is interesting to note that except for a constant factor, the Wronskian of two solutions ofL[y] = 0can be computed even if the solutionsy1andy2themselves are