New version page

WVU MATH 567 - Linear Systems

Documents in this Course
Load more
Upgrade to remove ads
Upgrade to remove ads
Unformatted text preview:

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

View Full Document
Download Linear Systems
Our administrator received your request to download this document. We will send you the file to your email shortly.
Loading Unlocking...

Join to view Linear Systems and access 3M+ class-specific study document.

We will never post anything without your permission.
Don't have an account?
Sign Up

Join to view Linear Systems 2 2 and access 3M+ class-specific study document.


By creating an account you agree to our Privacy Policy and Terms Of Use

Already a member?