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WVU MATH 567 - Linear Systems

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Linear systems of first order DE’sSystems of the form x′ Atx btExample:x1′ tx1− x2 costx2′ 2x1− etx2canbewrittenintheformx′t −12 −etxcost0, where xx1x2Note: Any scalar DE of order n can be rewritten as a system of n first order DE’s.Basic theory:Existence/uniqueness:An initial value problem includes an initial condition of the form xt0 x0. Initial valueproblems for linear systems of first order DE’s have a unique solution on the largestopen interval I containing t0for which the entries in Atand btare continuous.Homogenous system: x′ AtxFor such equations, we have superposition of solutions: any linear combination ofsolutions is a solution. This is easy to see if we rewrite the equation asLx x′− Atx 0and note that L is linear, namely Lū  v Lū LvandLaū aLū.To obtain the general solution of x′ Atxon an interval I whereexistence/uniqueness holds, we need only have a family of solutions that can satisfyany set of initial conditions at some t  t0contained in I. If we are working in Rn(i.e.we have n first order DE’s) then we are looking for n solutions of x′ Atxgiven byx1t,...,xntso that xt c1x1t...cnxntis the general solution. Such a set ofsolutions is called a fundamental set of solutions. This will be the case if the vectorsx1t0,...,xnt0are linearly independent at some t  t0since in that case,x0 xt0 c1x1t0...cnxnt0will have a solution for any chosen x0. Note thatindependence of x1t,...,xntat t  t0implies independence throughout the intervalI.Non-homogeous system: x′ Atx bthas the general solution x xh xpwherexhis the general solution of x′ Atxand xpis any particular solution ofx′ Atx bt. We can actually write a simple formula for xp, which we do a bitfurther down.Fundamental matrix: If we take a fundamental set of solutions of x′ Atxand putthem in a matrix tx1t... xntthen such a matrix is called a fundamentalmatrix of the DE x′ Atx. Then linear combinations of the solutions1xt c1x1t...cnxntcanbewrittenxt tc. An initial condition xt0 x0then results in x0 t0cand c −1t0x0and xt t−1t0x0gives a formalway of writing the solution. Note that the matrix t t−1t0is also afundamental matrix (each column is a linear combination of solutions and so is asolution and its columns can clearly satisfy any initial condition at t  t0 and talsohas the nice property that xttx0is the solution of x′ Atx, xt0 x0. This isthe same as the property that t0 I. A fundamental matrix tsuch that t0 Iis said to be "normalized at t  t0". We have just observed that if tis anyfundamental matrix, then t−1t0is a fundamental matrix normalized at t  t0.Variation of parameters formula: Once we have the general solution of thehomogenous system, we can write a formula for a particular solution ofx′ Atx bt: xp t−1tbtdt is a particular solution, or morespecifically, xp tt0t−1sbsds is the particular solution satisfying xpt0 0.Then the solution of the initial value problem x′ Atx bt, xt0 x0is given byx t−1t0x0 tt0t−1sbsds. Notice how much "simpler" this is than our oldvariation of parameters formula - the general idea is clearly exhibited, while thealgebraic details are hidden in the inverse notation.Constant coefficient coefficient homogeneous linear systems:We cannot hope to solve general equations x′ Atxin terms of formulas, althoughwe have demonstrated some nice properties of the solutions of such linear systems.After all, even a simple equation such as Airy’s equation, y′′ ty, or in matrix formx′01t 0xwhere x1 y , x2 y′, has no solution that can be written in terms ofelementary functions. We concentrate here on the case where the matrix A isconstant. Such types of equations are important, and arise naturally, when nonlinearautonomous DE’s are linearized - autonomous means that t does not explicitly appearin the equation.Solutions of x′ Ax: From here on, we assume that A is constant.x′ Axhas a solution of the form xt etv, where vis a constant vector, if andonly if  is an eigenvalue of A and vis a corresponding eigenvector. This can be seenby plugging in: x′t etv Ax etAvif and only if Av v.The case of pure exponential solutions - n independent eigenvectors:If the matrix A has n independent eigenvectors (as would be the case, for instance, ifthe eigenvalues were distinct roots of the characteristic equation) then we obtain nsolutions xjt ejtvj, j  1,..,n that are a fundamental set of solutions since at t0 0the vectors xj0 vj, j  1,..,n are linearly independent by assumption. In that case,"we are done", in the sense that we have the general solution of the formxt c1e1tv1 c2e2tv2...cnentvn. A fundamental matrix  can be composed fromthese solutions, and can be written in the form t Vetwhere V v1...vnis the2matrix of eigenvectors and etrepresents the diagonal matrixe1t 00  entwiththe terms ejtdown the diagonal. A fundamental matrix of solutions normalized at t  0can then be written, since 0 V, as t t−10 VetV−1.The matrix exponential:In the case of one equation, the DE x′ ax has solution x  eatx0. Can somethinglike this work in the matrix case? If A is an nxnconstant matrix, we want to define eAtin such a way


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