Transcription of Linear Systems of Differential Equations
1 LinearSystems ofDifferentialEquationsMath 240 First orderlinear systemsSolutionsBeyond firstorder systemsLinear Systems of Differential EquationsMath 240 Calculus IIIS ummer 2013, Session IIMonday, July 29, 2013 LinearSystems ofDifferentialEquationsMath 240 First orderlinear systemsSolutionsBeyond firstorder systemsAgenda1. First order Linear systemsSolutions to vector Differential equationsBeyond first order systemsLinearSystems ofDifferentialEquationsMath 240 First orderlinear systemsSolutionsBeyond firstorder systemsFirst order Linear systemsDefinitionAfirst order system of Differential equationsis of the formx (t) =A(t)x(t) +b(t),whereA(t)is ann nmatrix function andx(t)andb(t)aren-vector called avector Linear systemx 1(t) = cos(t)x1(t) sin(t)x2(t) +e tx 2(t) = sin(t)x1(t) + cos(t)x2(t)
2 E tcan also be written as the vector Differential equationx (t) =A(t)x(t) +b(t)whereA(t) =[cos(t) sin(t)sin(t) cos(t)],x(t) =[x1(t)x2(t)],andb(t) =[e t e t].LinearSystems ofDifferentialEquationsMath 240 First orderlinear systemsSolutionsBeyond firstorder systemsThe vector spaceVn(I)Asolutionto a vector Differential equation will be an elementof the vector spaceVn(I)consisting of columnn-vectorfunctions defined on the (t),x2(t), .. ,xn(t) Vn(I). TheWronskianofthese vectors isW[x1, .. ,xn](t) = | ||x1(t)x2(t) xn(t)| ||.
3 TheoremIfW[x1, .. ,xn](t)is nonzero for at least onet I, then{x1(t), .. ,xn(t)}is a linearly independent subset ofVn(I).LinearSystems ofDifferentialEquationsMath 240 First orderlinear systemsSolutionsBeyond firstorder systemsSolutions to homogeneous Linear systemsAs with Linear Systems , a homogeneous Linear system ofdifferential Equations is one in whichb(t) = (t)is ann nmatrix function that is continuous on theintervalI, then the set of all solutions tox (t) =A(t)x(t)is asubspace ofVn(I)of to ofdim =nlater, if there s ofDifferentialEquationsMath 240 First orderlinear systemsSolutionsBeyond firstorder systemsThe general solution: homogeneous caseIf the solution set is a vector space of dimensionn, it has set{x1,x2.}
4 ,xn}ofnsolutions tox =Axthat islinearly independent onIis called afundamental set solution may be written in the formx(t) =c1x1(t) +c2x2(t) + +cnxn(t),which is called thegeneral (t)is ann nmatrix function that is continuous on anintervalI, and{x1,x2, .. ,xn}is a linearly independent set ofsolutions tox =AxonI, thenW[x1,x2, .. ,xn](t)6= 0for everyt ofDifferentialEquationsMath 240 First orderlinear systemsSolutionsBeyond firstorder systemsThe general solution: nonhomogeneous caseThe case of nonhomogeneous Systems is also (t)is ann nmatrix function continuous on anintervalIand{x1.
5 ,xn}is a fundamental set of solutions tothe equationx =Ax. Ifx=xp(t)is any particular solution tothe nonhomogeneous vector Differential equationx (t) =A(t)x(t) +b(t)onI, then every solution to this equation onIis in the form ofthegeneral solutionx (t) = c1x1(t) +c2x2(t) + +cnxn(t)+xp(t),=xc(t)+xp(t)wherexp(t)is any particular two pieces of the general solution are theparticularsolution,xp(t), and thecomplementary solution,xc(t).LinearSystems ofDifferentialEquationsMath 240 First orderlinear systemsSolutionsBeyond firstorder systemsInitial value problemsSometimes, we are interested in one particular solution to avector Differential value problemconsists of a vector differentialequationx (t) =A(t)x(t) +b(t)and aninitial conditionx(t0) =x0with known, fixed values fort0 Randx0 (t)andb(t)
6 Are continuous on an intervalI, the aboveinitial value problem has a unique solution ofDifferentialEquationsMath 240 First orderlinear systemsSolutionsBeyond firstorder systemsTurning higher order Linear Systems into first orderAren t we a little limited if all we can solve are first orderdifferential Equations ? the linearsecondorder systemx (t) 4y(t) =et,y (t) +t2x (t) = new variablesx1(t) =x(t), x2(t) =x (t), x3(t) =y(t), x4(t) =y (t).Then the above Equations can be replaced withx 2(t) 4x3(t) =et,x 4(t) +t2x2(t) = sint,and we must supplement them with the equationsx 1(t) =x2(t), x 3(t) =x4(t).
