Homogeneous Linear Systems and Their General Solutions

37 Homogeneous Linear Systems and Their General Solutions We are now going to restrict our attention further to the standard first-order systems of differential equations that are linear, with particular attention being given to developing the theory for solving those linear systems that are also homogeneous Fortunately, this theory is very similar to that for single linear differential equations developed in chapters 3 and 4 In fact, to some extent, our discussion will be guided by what we already know about general solutions to N th -order linear differential equations You should also expect to see significant use of a few results from basic linear algebra Will we finally actually solve a few systems in this chapter? No, not really, but we will need the theory developed here when we finally do start solving systems in the next chapter 37 Basic Terminology and Notation Basic Definitions A standard first-order N N system x x 2 = x N f (t, x, x 2,, x N ) f 2 (t, x, x 2,, x N ) f N (t, x, x 2,, x N ) is said to be linear if and only if each component function f k can be written as f k (t, x, x 2,, x N ) = p k x + p k2 x 2 + + p k N x N + g k where the p k j s and g k s are either constants or functions of t only If, in addition, all the g k s are zero, so that our system looks like x p x + p 2 x 2 + + p N x N x 2 = p 2 x + p 22 x 2 + + p 2N x N, (37) x N p N x + p N2 x 2 + + p N N x N

Chapter & Page: 37 2 Homogeneous Linear Systems and Their General Solutions then we say our linear system is homogeneous If, on the other hand, we have x p x + p 2 x 2 + + p N x N g x 2 = p 2 x + p 22 x 2 + + p 2N x N + g 2 x N p N x + p N2 x 2 + + p N N x N g N (372) with one or more of the g k s being nonzero, then we say that the linear system is nonhomogeneous Matrix/Vector Notation for Linear Systems Letting P = p p 2 p N p 2 p 22 p 2N p N p N2 p N N g g 2 g N and g =, and using the standard rules of matrix multiplication, p p 2 p N x p x + p 2 x 2 + + p N x N p 2 p 22 p 2N x 2 Px = = p 2 x + p 22 x 2 + + p 2N x N, p N p N2 p N N x N p N x + p N2 x 2 + + p N N x N we can rewrite homogeneous linear system (37) and nonhomogeneous linear system (372), respectively, as and x = Px (37 ) x = Px + g (372 ) Keep in mind that the components of both P and g may be functions of t, but not of any of the x k s Also note that the system is automatically a regular autonomous system if all the components of P and g are constants As with vectors, we will refer to any matrix P as being a constant matrix or a matrix-valued function (on some interval) according to whether the components of P are all constants or can be functions (on the given interval) If, in addition, all the components of P are continuous functions on some interval, then we will say P is a continuous matrix-valued function on that interval Let us also agree that (unless otherwise indicated) all components of our matrices and vectors are real-valued If it seems particularly relevant, we ll explicitly say that a given matrix is real or real-valued to indicate that its components are real values or real-valued functions Expanding on our conventions from the previous chapter, we will, as much as possible, use bold-faced capital letters to denote generic N N matrices, with the corresponding lower-case, nonbold letters, suitably subscripted, to denote the corresponding components of the matrix So if we have a matrix A, then (unless otherwise indicated) a a 2 a N a 2 a 22 a 2N A = a N a N2 a N N

Basic Terminology and Notation Chapter & Page: 37 3 Let us also agree that, when discussing a standard N N linear system, every vector and matrix under discussion consisting of a single row or column has N components, and that every matrix under discussion not consisting of a single row or column is a N N matrix Sets of Vectors Keep in mind that the x in the above discussion denotes a vector-valued function on some interval, x(t) = [ x (t), x 2 (t),, x N (t) ] T Often, in what follows, we will have several such vector-valued functions When we do, we may use superscripts to distinguish the different vector-valued functions; that is, we will write the set of vector-valued functions as either { x, x 2,, x M } or { } x (t), x 2 (t),, x M (t), with x (t) = x (t) x 2 (t) xn (t), x 2 (t) = x 2 (t) x 2 2(t) xn 2 (t), and x M (t) = x M (t) x 2 M(t) xn M(t)! Example 37: Consider the system [ x ] = y [ ] x + 2y x 2y This is easily seen to be a homogeneous linear 2 2 system of differential equations, and since [ ] [ ][ ] x + 2y 2 x =, x 2y 2 y we see that we can rewrite it as either [ x ] = or even as where x = [ ] x y y [ ][ 2 x 2 y x = Px and P = ] [ ] 2 2 In this case P is a constant matrix and, hence, is automatically a continuous 2 2 matrix-valued function over the interval (, ) It is easily verified (see example 32 and exercise 34) that one pair of solutions { x, x 2} to the above system is given by [ ] [ ] e x 3t 2e (t) = and x 2 4t (t) =, e 3t e 4t which we can write more simply as [ ] [ ] 2 x (t) = e 3t and x 2 (t) = e 4t

Chapter & Page: 37 4 Homogeneous Linear Systems and Their General Solutions 372 More Terminology and Some Basic Results Let us now focus on determining the basic nature of the general solutions to a homogeneous N N linear system x = Px where, for some positive integer N, p (t) p 2 (t) p N (t) p 2 (t) p 22 (t) p 2N (t) P = P(t) = (373) p N (t) p N2 (t) p N N (t) is a continuous N N matrix-valued function on some interval (α, β) Our goal is to extend the basic notions and results developed back in chapters 3 and 4 of the text for single N th -order homogeneous linear equations The main results of our development are summarized in theorem 379 on page 37 2 You can go ahead and look at this theorem Even though some of the terminology has not yet been formally defined in the context of systems, the terminology and theorem are so similar to that already developed in chapter 3 of the text that you will probably be able to understand the gist of the theorem That said, if we are to intelligently use theorem 379, we need to fully understand and verify the claims in this theorem We will develop that understanding and verify those claims, piece by piece, in this section This will include developing some of the concepts and terminology used in that theorem Many of these will be concepts and terms that you should recall from your study of linear algebra By the way, in the following, we will also be referring to constants, vectors and, maybe, constant vectors Just to be clear, when we refer to something as being just a constant (not constant vector), then we mean that something is a single real number And if we refer to something as just a vector or constant vector then that something is a column vector whose N components are constants So a is a vector means a = [a, a 2,, a N ] T with each a k being some single real number Immediate Results on Existence and Uniqueness Let us start with the basic theorem on the existence and uniqueness of solutions to linear systems of differential equations Theorem 37 (existence and uniqueness for linear systems) Assume P is a continuous N N matrix-valued function over the interval (α, β) and g is a continuous vector-valued function over (α, β), and let t and a be, respectively, a point in (α, β) and a constant vector Then the initial-value problem x = Px + g with x(t ) = a, has exactly one solution over the interval (α, β) Moreover, this solution and its derivative are continuous over that interval The above theorem is actually a corollary of an earlier theorem on the existence and uniqueness of solutions to systems with sufficiently continuous component functions The verification is left to you It is worth recalling that the set of all such column vectors with N components (with the standard definitions of vector addition and scalar multiplication) is an N-dimensional vector space

More Terminology and Some Basic Results Chapter & Page: 37? Exercise 37: Verify the claims in theorem 37 by verifying that the component functions of the system in the theorem 37 satisfy the requirements given in theorem 32 on page 76 of the published text, and then applying theorem 32 By letting g = in the above theorem, we get the result we will actually use in this chapter: Lemma 372 (existence and uniqueness for homogeneous linear systems) Assume P is a continuous N N matrix-valued function over the interval (α, β), and let t and a be, respectively, a point in the interval (α, β) and a constant vector Then the initial-value problem x = Px with x(t ) = a, has exactly one solution over the interval (α, β) Moreover, this solution and its derivative are continuous over that interval By the way, this a good place to observe that the constant zero vector-valued function, x(t) = for all t, is always a solution to any given homogeneous linear system x = Px Consequently, the origin (,,,) is always a critical point a critical point for the given system This is analogous to the fact that the zero function is always a solution to any given single homogeneous linear differential equation This will be particularly significant in a few chapters Linear Combinations and the Principle of Superposition Recall that a linear combination of x k s from any finite set {x, x 2,, x M} of either vectors or vector-valued functions is any expression of the form c x + c 2 x 2 + + c M x M where the c k s are constants Keep in mind that, if the x k s are vector-valued functions on the interval (α, β), then x = c x + c 2 x 2 + + c M x M means x(t) = c x (t) + c 2 x 2 (t) + + c M x M (t) for α < t < β Now suppose we have a linear combination c x + c 2 x 2 + + c M x M in which each of these x j s is a solution to our linear system of differential equations; that is, dx j dt = Px j for j =, 2,, M Because of the linearity of differentiation and matrix multiplication, we then have d [c x + c 2 x 2 + + c M x M] dx = c dt dt + c 2 dx 2 dt + + c M dx M = c Px + c 2 Px 2 + + c M Px M [ = P c x + c 2 x 2 + + c M x M] dt

Chapter & Page: 37 6 Homogeneous Linear Systems and Their General Solutions Cutting out the middle yields the systems version of the superposition principle: Lemma 373 (principle of superposition for systems) If x, x 2, and x M are all solutions to a homogeneous linear system x = Px, then so is any linear combination of these x k s Observe that, if {x, x 2,, x M } is a set of solutions to our system x = Px, and x is any single solution equaling some linear combination of the x k s at one single point t in (α, β), then x(t ) = c x (t ) + c 2 x 2 (t ) + + c M x M (t ), (374) x and c x + c 2 x 2 + + c M x M are both solutions to x = Px satisfying the same initial condition at t But lemma 372 tells us that there is only one solution to this initial-value problem Hence, x and this linear combination must be the same That is, x(t) = c x (t) + c 2 x 2 (t) + + c M x M (t) for every value t in (α, β) (37) This, along with the obvious fact that equation (37) implies equation (374), gives us our next lemma Lemma 374 Let { x, x 2,, x M} be any set of solutions to x = Px, where P is a continuous N N matrixvalued function on the interval (α, β) Also let {c, c 2,, c M } be a set of constants, and let t be a point in the interval (α, β) Then, for any solution x to x = Px, if and only if x(t ) = c x (t ) + c 2 x 2 (t ) + + c M x M (t ) for one value t in (α, β) x(t) = c x (t) + c 2 x 2 (t) + + c M x M (t) for every value t in (α, β) An application using the above lemmas is now in order It will give you an idea of where we are heading! Example 372: We already know from exercise 37 that [ ] x (t) = e 3t and x 2 (t) = [ ] 2 e 4t are both solutions (over (, ) ) to x = Px with P = [ ] 2 2 The principle of superposition now assures us that, for any pair c and c 2 of constants, the linear combination [ ] [ ] 2 c x (t) + c 2 x 2 (t) = c e 3t + c 2 e 4t is also a solution to our homogeneous system

More Terminology and Some Basic Results Chapter & Page: 37 7 The obvious question now is whether every solution is given by a linear combination of x and x 2 To answer that, let x(t) = [x(t), y(t)] T be any single solution to x = Px, and consider the problem of finding constants c and c 2 such that x(t) = c x (t) + c 2 x 2 (t) for < t < According to our last lemma, this problem is completely equivalent to the problem of finding constants c and c 2 such that x(t ) = c x (t ) + c 2 x 2 (t ) for some t in (, ) Letting t =, and using the formulas for x and x 2, the last equation becomes [ ] [ ] [ ] x() 2 = c e 3 + c 2 e 4, y() which we can rewrite as pair of linear algebraic equations, x() = c 2c 2 y() = c + c 2 But you can easily solve this algebraic system and verify that, for each choice of x() and y(), the one and only one solution (c, c 2 ) to this system is given by c = 7 [y() x()] and c 2 = [6x() + y()] 7 Thus, using these values for c and c 2, we have x(t) = c x (t) + c 2 x 2 (t) for < t < So, at least for the system of differential equations being considered here, the answer to the question of whether every solution is given by a linear combination of x and x 2 is yes The above shows that, given any solution x, we can find one (and only one) corresponding pair of constants (c, c 2 ) such that [ ] [ ] 2 x(t) = c x (t) + c 2 x 2 (t) = c e 3t + c 2 e 4t In other words, the above expression is a general solution to our system x = Px As suggested in the above example, our goal is to show that, for any given P, every solution x to x = Px can be written as a linear combination of solutions from some fundamental set of solutions, { x, x 2,, x M } Moreover, as illustrated in the above example, we can use lemma 374 us to convert the problem of finding that linear combination x(t) = c x (t) + c 2 x 2 (t) + + c M x M (t) for α < t < β to the problem of finding constants c, c 2, and c M such that x(t ) = c x (t ) + c 2 x 2 (t ) + + c M x M (t )

Chapter & Page: 37 8 Homogeneous Linear Systems and Their General Solutions for a single t But remember that another lemma, lemma 372, assures us that there is a solution x to our system of differential equations satisfying x(t ) = a for each vector a and each t in (α, β) Combining this fact with the results from lemma 374 gives our next lemma Lemma 37 Assume P be a continuous N N matrix-valued function on the interval (α, β) Let t be a point in this interval, and let {x, x 2,, x M} be any set of solutions to x = Px Then every solution x to x = Px can be written as a linear combination of the x k s if and only if every vector a can be written as a linear combination of vectors from the set { } x (t ), x 2 (t ),, x M (t ) This lemma, along with a similar lemma concerning linear independence, will play a major role in our final derivations So let s now bring back the basic notion of linear (in)dependence Linear Independence Let { x, x 2,, x M } be either a set of vectors or a set of vector-valued functions on (α, β) We say that this set is linearly independent if and only if none these x k s can be written as as linear combination of the other x k s Otherwise, we say this set is linearly dependent; that is, the set is linearly dependent if and only if at least one these x k s can be written as as linear combination of the other x k s Two quick observations: Any constant multiple of a single x k is a (very simple) linear combination of that x k In particular, since = x k, any set containing the zero vector or the zero vector-valued function is automatically linearly dependent 2 If we just have a pair { x, x 2}, the concept of linear independence simplifies to the pair being linearly independent if and only if neither x nor x 2 is a constant multiple of the other! Example 373: where Consider the set { x, x 2} of vector-valued functions from the last example, x (t) = [ ] e 3t and x 2 (t) = [ ] 2 e 4t Clearly, there is no constant C such that either [ ] [ ] 2 e 3t = C e 4t for < t < or [ ] [ ] 2 e 4t = C e 3t for < t < So this set of two vector-valued functions is linearly independent Similarly, consider the set of vectors { b, b 2} given by the above vector-valued functions at t =, b = [ ] e 3 = [ ] and b 2 = [ 2 ] e 4 = [ ] 2

More Terminology and Some Basic Results Chapter & Page: 37 9 Again, it should be clear that there is no constant C such that either [ ] [ ] 2 = C or [ ] 2 = C [ ] So this set of two vectors is linearly independent Now suppose { x, x 2,, x M } is a set of solutions over (α, β) to our system x = Px, and let t be in (α, β) Lemma 374 tells us that any one solution x j is a linear combination of the other x k s if and only if the corresponding vector x j (t ) is a linear combination of the other x k (t ) s This observation is worth writing down as a lemma in terms of linear independence Lemma 376 Assume P is a continuous N N matrix-valued function on the interval (α, β) Let t be a point in this interval, and let { x, x 2,, x M } be any set of M solutions to x = Px Then this set is a linearly independent set of vector-valued functions if and only if { } x (t ), x 2 (t ),, x M (t ) is a linearly independent set of vectors Compare the above lemma with lemma 37 Both will play a major role in the following It will also be helpful to recall a test for linear independence that you should recall from your study of linear algebra 2 Lemma 377 (a basic test for linear independence) A set { x, x 2,, x M} of vectors or vector-valued functions is linearly independent if and only if the only choice of constants c, c 2, and c M such that is c x + c 2 x 2 + + c M x M = c = c 2 = = c M = then Observe that if we have two linear combinations of the same x k s equaling the same a, a = c x + c 2 x 2 + + c M x M and a = C x + C 2 x 2 + + C M x M, (c C )x + (c 2 C 2 )x 2 + + (c M C M)x M = a a = From this, you should have no problem in verifying that the above test for linear independence is equivalent to the following test 2 If you don t recall this test, see exercise 37 at the end of the chapter

Chapter & Page: 37 Homogeneous Linear Systems and Their General Solutions Lemma 378 (alternative test for linear independence) Let { x, x 2,,x M} be a set of vectors or vector-valued functions This set is linearly independent if and only if, for each a that can be written as a linear combination of the x k s, there is only one choice of constants c, c 2, and c M such that a = c x + c 2 x 2 + + c M x N Fundamental Sets of Solutions Basic Definition We now define a fundamental set of solutions for x = Px to be any linearly independent set of solutions to x = Px { x, x 2,, x M } such that every solution to x = Px can be written as a linear combination of the x j s in this set Note that, if the above is a fundamental set of solutions for x = Px, then x = c x + c 2 x 2 + + c N x M (with the c k s being arbitrary constants) is a general solution for x = Px Describing Fundamental Sets of Solutions It turns out that there are several other ways to describe fundamental sets To see this, let { } X = x, x 2,, x M be a set of solutions to x = Px where, as usual, P is a continuous N N matrix-valued function on an interval (α, β) Take any point t in the interval, and let { } B = b, b 2,, b M be the set of vectors given by b k = x k (t ) for k =, 2,, M From our basic definition of a fundamental set of solutions we know: The set X is a fundamental set of solutions to x = Px if and only if X is a linearly independent set of vector-valued functions such that any solution to x = Px can be written as a linear combination of the x k s From lemmas 37 and 376, we know this last statement is completely equivalent to: The set X is a fundamental set of solutions to x = Px if and only if B is a linearly independent set of vectors such that any vector can be written as a linear combination of the b k s Throwing in lemma 378 we get another equivalent statement: The set X is a fundamental set of solutions to x = Px if and only if, for each vector a = [a, a 2,,a N ] T, there is one and only one choice of constants c, c 2, and c M such that a = c b + c 2 b 2 + + c M b M

More Terminology and Some Basic Results Chapter & Page: 37 At this point, you probably realize that the last two statements are saying that the set of x k s is a fundamental set of solutions if and only if the set of b k s is a basis for the vector space of all column vectors with N components, and, from linear algebra, we know that M, the number of vectors in the set B must equal N the number of components in each column vector Moreover, from linear algebra, we know that any set of N linearly independent vectors will be a basis for this space of column vectors 3 So either of the last two statements about X can be rephrased as The set X is a fundamental set of solutions to x = Px if and only if M = N and B is a linearly independent set of vectors Applying lemma 376 once again with the last yields: The set X is a fundamental set of solutions to x = Px if and only if M = N and X is linearly independent All of the above could be considered pieces of one big lemma Rather than state that lemma here, we will summarize the most relevant pieces in a major theorem in the next section, after making a final observation on the existence of fundamental solution sets Existence of Fundamental Sets of Solutions Let us observe that fundamental sets of solutions clearly do exist After all, no matter what N is, we can always find a linearly independent set of N vectors with N components, { b, b 2,, b N } For example, if N = 3 we can use b =, b 2 = and b 3 =, if N = 4 we can use b =, b 2 =, b 3 = and b 4 =, and so on Lemma 372 then assures us that, for any point t in (α, β) and every b k, there is a solution x k to x = Px satisfying x k (t ) = b k As noted in the last subsection above, it then follows that { x, x 2,, x N } is a fundamental set of solutions to our system of differential equations 3 An alternative derivation not using basis of the fact that M = N is given in section 374

Chapter & Page: 37 2 Homogeneous Linear Systems and Their General Solutions 373 The Main Result on General Solutions to Linear Systems Looking back over the discussion on fundamental sets of solutions in the last section, you will see that we have verified the following major theorem on general solutions to linear systems of differential equations Theorem 379 (general solutions to homogenous systems) Let P be a continuous N N matrix-valued function on an interval (α, β), and consider the system of differential equations x = Px Then all the following statements hold: Fundamental sets of solutions over (α, β) for this system exist 2 Every fundamental set of solutions consists of exactly N solutions 3 If { x, x 2,, x N} is any linearly independent set of N solutions to x = Px on (α, β), then (a) {x, x 2,, x N } is a fundamental set of solutions for x = Px on (α, β) (b) A general solution to x = Px on (α, β) is given by x(t) = c x (t) + c 2 x 2 (t) + + c N x N (t) (c) where c, c 2, and c N are arbitrary constants Given any single point t in (α, β) and any constant vector a, there is exactly one ordered set of constants {c, c 2,,c N } such that x(t) = c x (t) + c 2 x 2 (t) + + c N x N (t) satisfies the initial condition x(t ) = a This theorem is the systems analog of theorem 3 on page 272 concerning general solutions to single N th -order homogeneous linear differential equations In fact, theorem 3 can be considered a corollary to the above We verify that in section 376 374 Wronskians and Identifying Fundamental Sets As illustrated in the previous examples, determining whether a set of solutions is a fundamental set for our problem x = Px is fairly easy when P is 2 2 Our goal now is to come up with a method for identifying a fundamental set of solutions that be easily applied when P is N N even when N > 2 Let us start by assuming we have a set { x, x 2,, x M }

Wronskians and Identifying Fundamental Sets Chapter & Page: 37 3 of vector-valued functions on the interval (α, β), each with N components, x (t) = x (t) x 2 (t) xn (t), x 2 (t) = x 2 (t) x 2 2(t) xn 2 (t), and x M (t) = x M (t) x 2 M(t) xn M(t) For the moment, we need not assume the x k s are solutions to our N N system of differential equations, nor will we assume N = M A Matrix/Vector Formula for Linear Combinations Observe: x c x + c 2 x 2 + + c M x M 2 x 2 2 x M = c + c 2 + + c M 2 x x 2 x M x N x 2 N x M N = = x c + x 2c 2 + + x Mc M x2 c + x2 2c 2 + + x2 Mc M xn c + xn 2 c 2 + + xn Mc M x x 2 x M x2 x2 2 x M 2 xn xn 2 xn M c c 2 c M That is, for α < t < β, c x (t) + c 2 x 2 (t) + + c M x M (t) = [X(t)]c where X(t) = x (t) x2 (t) x M(t) x2 (t) x2 2 (t) x 2 M(t) xn (t) x2 N (t) x N M(t) and c = c c 2 c M The above N M matrix-valued function X will be important to us In general, we ll simply call it the matrix whose k th column is given by x k! Example 374: The matrix whose k th column is given by x k when [ ] [ ] e x 3t 2e (t) = and x 2 4t (t) = e 3t e 4t

Chapter & Page: 37 4 Homogeneous Linear Systems and Their General Solutions is [ ] e 3t 2e 4t X(t) = e 3t e 4t Observe that, indeed, [ ] e 3t 2e 4t [c ] [X(t)]c = e 3t e 4t c 2 = [ ] c e 3t + c 2 ( 2)e 4t [ ] e 3t = c c e 3t + c 2 e 4t e 3t [ ] 2e 4t + c 2 e 4t = c x (t) + c 2 x 2 (t) Deriving that Simple Test Now assume these x k s are solutions to our system x = Px, and let t be any single value in (α, β) From lemmas 37, 376 and 378, we know (as noted on page 37 using slightly different notation) that: The set { x, x 2,,x M} is a fundamental set of solutions to x = Px if and only if, for each vector a = [a, a 2,, a N ] T, there is one and only one choice of constants c, c 2, and c M such that c x (t ) + c 2 x 2 (t ) + + c M x M (t ) = a (376) However, from the observations made just before our last example, we know that equation (376) is equivalent to the algebraic system of N equations and M unknowns x (t )c + x 2 (t )c 2 + + x M (t )c M = a x 2 (t )c + x 2 2 (t )c 2 + + x M 2 (t )c M = a 2, (377) x N (t )c + x 2 N (t )c 2 + + x M N (t )c M = a N which can also be written as the matrix/vector equation [X(t )] c = a (378) where c = [c, c 2,,c M ] T and X(t) is the N M matrix whose k th column is given by x k (t) But solving either algebraic system (377) or matrix/vector equation (378) is a classic problem in linear algebra, and from linear algebra we know there is one and only one solution c for each a if and only if M = N and X(t ) is invertible If these two conditions are both satisfied, then c can be determined from each a by c = [X(t )] a where [X(t )] is the inverse of matrix X(t ) (In practice, though, a row reduction method may be a more efficient way to find c )

Wronskians and Identifying Fundamental Sets Chapter & Page: 37 Now, to make life even easier, recall that there is a relatively simple test for determining if a given square matrix M is invertible 4 based on the matrix s determinant, det(m) ; namely, M is invertible det(m) = Thus, our set of M solutions is a fundamental set of solutions if and only if M = N and det(x(t )) = Wronskians and Identifying Fundamental Sets The last line above gives us a useful test for determining if a given set of solutions is a fundamental set of solutions It also gives the author an excuse for introducing additional terminology concerning any set {x, x 2,, x N} of N vector-valued functions on an interval (α, β), with each x k Wronskian, W, of this set is the function on (α, β) given by having N components The W(t) = det(x(t)) where X is the matrix whose k th column is given by x k Using the Wronskian, we can now properly state the test we have just derived above Theorem 37 (Identifying Fundamental Sets of Solutions) Let { x, x 2,, x M} be a set of M solutions to x = Px, with P being a continuous N N matrix-valued function on an interval (α, β) Then this set is a fundamental set of solutions for x = Px if and only if both of the following hold: M = N 2 For any single t in (α, β), W(t ) =, where W is the Wronskian of { x, x 2,, x M}! Example 37: are x = It is not hard to verify that three solutions (on (, ) to 2 2 4 x = Px with P = 2 3 6 8 2 3 e 2t, x 2 = 3 e 2t and x 3 = e 2t 3 3 4 More terminology you should recall: M is singular M is not invertible M is nonsingular M is invertible

Chapter & Page: 37 6 Homogeneous Linear Systems and Their General Solutions The corresponding matrix whose k th columm given by x k is e 2t 2e 2t 3e 2t X(t) = e 2t 3e 2t e 2t 3e 2t e 2t 3e 2t and the Wronskian is W(t) = det(x(t)) = e 2t 2e 2t 3e 2t e 2t 3e 2t e 2t 3e 2t e 2t 3e 2t Computing out this determinant is not difficult, but not necessary All we need is to compute is W(t ) for some convenient value t, say t =, e 2 2e 2 3e 2 W() = det(x()) = e 2 3e 2 e 2 3e 2 e 2 3e 2 = 2 3 3 3 3 = 3 3 2 3 3 + 3 3 3 = [9 + ] 2[3 3] + 3[ 9] = 2 Since W() =, the above theorem tells us that the set { x, x 2, x 3} is a fundamental set of solutions for the above system of differential equations And that means 2 3 x(t) = c e 2t + c 2 3 e 2t + c 3 e 2t 3 3 is a general solution to the 3 3 system x = Px being considered here By the way, the fact that we can choose t arbitrarily in (α, β) tells us that whether W(t ) is zero or not is totally independent of the choice of t That gives us the following corollary Corollary 37 Assume P is a continuous N N matrix-valued function on an interval (α, β), and let W be the Wronskian of a set of N solutions to x = Px Then if and only if W(t ) = for one value t in (α, β) W(t) = for every value t in (α, β)

General Solutions for a Single N th -order Linear Differential Equation Chapter & Page: 37 7 37 Fundamental Matrices In the last section, we introduced the matrix-valued function X whose k th column is given by the k th vector-valued function in a set { x, x 2,, x N} In the future, we will refer to X as a fundamental matrix for x = Px if and only if the above set is a fundamental set of solutions for x = Px Fundamental matrices will play a role in some of our later discussions! Example 376: In example 37, just above, we considered the problem 2 2 4 x = Px with P = 2 3, and saw that the set {x, x 2, x 3 } with 2 x = e 2t, x 2 = 3 e 2t and x 3 = 3 6 8 3 e 2t 3 is a fundamental set of solutions to the given problem x = Px Hence, the matrix whose k th columm given by x k, e 2t 2e 2t 3e 2t X(t) = e 2t 3e 2t e 2t, 3e 2t e 2t 3e 2t is a fundamental matrix for this problem For future reference, let us note that the following lemma follows immediately from the discussion in the previous section: Lemma 372 If X is a fundamental matrix for x = Px where P is a continuous matrix-valued function on an interval (α, β), then X(t) is an invertible matrix for each t in (α, β) 376 General Solutions for a Single N th -order Linear Differential Equation In chapter 3 we presented two important theorems concerning single N th -order linear differential equations the main theorem on general solutions, theorem 3 on page 272, and a theorem The material in this section plays no role in later developments, and can be safely skipped by those more interested in learning more about systems of differential equations than in verifying theorems for single equations

Chapter & Page: 37 8 Homogeneous Linear Systems and Their General Solutions on using Wronskians to identify fundamental sets of solutions, theorem 36 on page 274 Those theorems were proven in that and the following chapter for the case where N = 2, but were not completely verified assuming N > 2 Using the results just derived in the chapter, we can now do so Let us see how The Basic Assumptions Throughout this section, we are concerned with a fairly general N th -order linear differential equation over an interval (α, β) a y (N) + a y (N ) + + a N 2 y + a N y + a N y = (379a) where each a k is a continuous function over (α, β) and with a (t) never being zero in that interval Solving for y (N), we see that this equation can also be written as y (N) = p N y + p N2 y + + p N N y (N ) (379b) where p N = a N a, p N2 = a N a, and p N N = a a Observe that the assumptions made on the a k s ensure that each of these p Nk s is a continuous function on (α, β) On occasion, we will also be interested in a solution to the above differential equation that satisfies some N th -order set of initial conditions y(t ) = a, y (t ) = a 2, y (t ) = a 3, and y (N ) (t ) = a N where t is some point in (α, β) and the A k s are real numbers (37) The Analysis As discussed in section 36 of the published text, we can convert equation (379b) to an equivalent standard N N system x = F(t, x) by introducing N new unknown functions x, x 2, and x N related to y and each other by x 3 = x 2 x = y, x 2 = x = y = y, and x N = x N = y (N ), and observing that, because of differential equation (379b), x N = y N = p N x + p N2 x 2 + + p N N x N This gives us the standard system x x 2 = x N x N x 2 x 3 x N p N x + p N2 x 2 + + p N N x N,

General Solutions for a Single N th -order Linear Differential Equation Chapter & Page: 37 9 which we immediately recognize as being a homogeneous linear system x = Px with P = p N p N2 p N3 p N4 p N N Moreover, since the p Nk s are continuous functions on (α, β), it should be clear that P is a continuous matrix-valued function on (α, β) In addition, because of the relations between the x k s and the derivatives of y, set (37) of initial conditions becomes the initial condition x(t ) = a where It should further be clear that: a = [a, a 2,,, a N ] T If y is a solution to differential equation (379a) which satisfies initial condition set (37), and x = [ y y y y (N )] T, then x is a solution to x = Px satisfying initial condition x(t ) = a 2 If x = [x, x 2,, x N ] T is a solution to x = Px which satisfies initial condition x(t ) = a, and y = x, then (a) x = [y, y, y,, y N ) ] T, and (b) y is a solution to differential equation (379a) satisfying initial condition (37) Thus, for each solution y to differential equation (37) there is a single corresponding solution x to x = Px Likewise, for each solution x to x = Px there is a corresponding solution y to differential equation (37) Moreover, these corresponding pairs are related by So now let x = [ y y y y (N )] T { y, y 2,, y M } and { x, x 2,, x M } be, respectively, a set of functions and vector-valued functions on (α, β) with each y k and x k related as just described, x k = [ y y k y k y k (N ) ] T Further assume that either all the y k s are solutions to differential equation (37) OR that all the x k s are solutions to x = Px Then the above assures us that, in fact, all the y k s are solutions to differential equation (37) AND that all the x k s are solutions to x = Px Moreover, you should have little trouble in verifying each of the following: Let y and x be a corresponding pair of solutions to differential equation (37) and x = Px (as described above), respectively, and let c, c 2, nad c M be some collection of constants Then y = c y + c 2 y 2 + + c M y M if and only if x = c x + c 2 x 2 + + c M x M

Chapter & Page: 37 2 Homogeneous Linear Systems and Their General Solutions 2 Let c, c 2, and c M be some collection of constants, and let and y = c y + c 2 y 2 + + c M y M x = c x + c 2 x 2 + + c M x M Then y satisfies the initial condition set (37) if and only if x(t ) = a 3 Every solution of differential equation (37) can be written as a linear combination of the y k s if and only if every solution of x = Px can be written as a linear combination of the x k s 4 The set of y k s is linearly independent if and only if the set of x k s is linearly independent The set of y k s is a fundamental set of solution to differential equation (37) if and only if the set of x k s is a fundamental set of solutions to x = Px? Exercise 372: Justify each of the five claims made above It should now be very clear why theorems 3 and 379 are so similar, and why it was claimed that theorem 3 (which we did not actually prove) can be derived from theorem 379 (which we did prove) Why don t you do that derivation yourself?? Exercise 373: Using the above results and theorem 379, verify the claims in theorem 3 Finally, let s deal with the claims made in chapter 3 concerning Wronskians Let { y, y 2,, y N } be any set of N solutions to differential equation (37), and let { x, x 2,, x N } be the corresponding set of solutions to x = Px with each y k and x k related by x k = [ y y k y k y k (N ) ] T Starting with the definition of Wronskian given in chapter 3, we see that y y 2 y 3 y N y y 2 y 3 y N Wronskian of {y, y 2,, y N } = y y 2 y 3 y N y (N ) y (N ) 2 y (N ) 3 y (N ) N x x 2 x N x2 x2 2 x N 2 = xn xn 2 xn N { = Wronskian of x, x 2,, x N}

Additional Exercises Chapter & Page: 37 2 Using this, it is a simple matter to show that the general results on Wronskians given in chapter 3 (theorem 36 on page 274) follow from the results concerning Wronskians proven in this chapter (theorem 37 and corollary 37) Additional Exercises 374 Rewrite each of the following linear systems of differential equations in matrix/vector form a x = 3x + y y = x + 7y b x = x 2 x 2 = x 3 x 3 = 4x + 3x 2 2x 3 c x = 2x y + 2z + 4 y = 2y 4z + z = 9x 3z + 6 d x = 2x 2 t 2 x 3 + sin(t) x 2 = (t + )x + tx 3 cos(t) x 3 = 3t 3 x 2 + t 37 Consider the two equations x M = C x + C 2 x 2 + + C M x M (37) and c x + c 2 x 2 + + c M x M = (372) where {x, x 2,,x M } is a set of vector-valued functions on an interval (α, β) a Using simple algebra, show that equation (37) holds for some constants C, C 2, and C M if and only if equation (372) holds for some constants c, c 2, and c M with c M = b Expand on the above and explain how it follows that at least one of the x k s must be a linear combination of the other x k s if and only if equation (372) holds with at least one of the c k s being nonzero c Finish proving lemma 377 on page 37 9 376 Consider the system x = y y = 4t 2 x + 3t y a Rewrite this system in matrix/vector form b What are the largest intervals over which we can be sure solutions to this system exist? c Verify that [ t x 2] (t) = 2t are both solutions to this system and x 2 (t) = [ t 2 ] ln t t( + 2 ln t )

Chapter & Page: 37 22 Homogeneous Linear Systems and Their General Solutions d Compute the Wronskian W(t) of the set of the above x k s at some convenient nonzero point t = t (part of this problem is to choose a convenient point) What does this value of W(t ) tell you? e Using the above, find the solution to the above system satisfying i x() = [, ] T ii x() = [, ] T 377 Consider the system x = x + 2y 2z y = 2x + 4y 2z z = a Rewrite this system in matrix/vector form 2x + 2y 4z b What is the largest interval over which we are sure solutions to this system exist? c Verify that x(t) =, y(t) = e 2t and z(t) = 2 e 2t 2 are all solutions to this system d Compute the Wronskian W(t) of the set {x, y, z} some convenient point t = t (choosing a convenient point is part of the problem), and verify that the above {x, y, z} is a fundamental set of solutions to the above system of differential equations 378 Four solutions to 2 x = x 2 are cos(2t) sin(2t) sin 2 (t) x (t) = sin(2t), x 2 (t) = cos(2t), x 3 (t) = sin(t) cos(t) cos(2t) sin(2t) cos 2 (t) and x 4 (t) = Given this, determine which of the following are fundamental sets of solutions to the given system: { a x, x 2} { b x, x 4} c {x, x 2, x 3} { d x, x 2, x 4} { e x, x 3, x 4} f {x, x 2, x 3, x 4} 379 Four solutions to x = 2 8 4 x 4

Additional Exercises Chapter & Page: 37 23 are and x (t) = 2 e 3t, x 2 (t) = e 3t, x 3 (t) = 2 x 4 (t) = 2 e t 3 4 e 3t 4 Given this, determine which of the following are fundamental sets of solutions to the given system: { a x, x 2} { b x, x 4} c {x, x 2, x 3} { d x, x 2, x 4} { e x, x 3, x 4} f {x, x 2, x 3, x 4} 37 Traditionally (ie, in most other texts), corollary 37 on page 37 6 is usually proven by showing that the Wronskian W of a set of N solutions to an N N system x = Px satisfies the differential equation W = [ p, + p 2,2 + + p N,N ] W, and then solving this differential equation and verifying that the solution is nonzero over the interval of interest if and only if it is nonzero at one point in the interval Do this yourself for the case where N = 2

Chapter & Page: 37 24 Homogeneous Linear Systems and Their General Solutions Some Answers to Some of the Exercises WARNING! Most of the following answers were prepared hastily and late at night They have not been properly proofread! Errors are likely! [ ] 4a x y = [ ][ 3 xy ] 7 [ x ] [ ][ x ] 4b x 2 = x 2 x 3 4 3 2 x 3 [ x ] [ ] 2 2 [ xy ] [ 4c x 4 ] 2 = 2 4 + x 3 9 3 z 6 [ x ] [ ] 2 t 2 [ xy ] [ sin(t) ] 4d x 2 = t+ t + cos(t) x 3 3t 3 z [ ] [ ] t 6a x [ y = xy ] 4t 2 3t 6b (, ) and (, ) 6d W() = = (Hence {x [, x 2 } is a fundamental set of solutions) 6e i x(t) = x (t) 2x 2 t (t) = 2 ] ( 2 ln t ) 4t ln t [ 6e ii x(t) = x 2 t (t) = 2 ] ln t t( + 2 ln t ) [ ] ] x [ xy ] 7a y = z [ 2 2 2 4 2 2 2 4 z 7b (, ) 7d W() = 8a It is not a fundamental set since the set is too small 8b It is not a fundamental set since the set is too small 8c It is a fundamental set there are three solutions in the set, and W() = 8d Is a fundamental set there are three solutions in the set, and W() = 8e Is not a fundamental set there are three solutions in the set, but W() = 8f Is not a fundamental set the set is too large 9a It is not a fundamental set the set is too small 9b It is not a fundamental set the set is too small 9c It is not a fundamental set there are three solutions in the set, but W() = 9d It is a fundamental set there are three solutions in the set, and W() = 9e It is a fundamental set there are three solutions in the set, and W() = 9f It is not a fundamental set the set is too large