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Advanced Engineering Mathematics 5 Linear Systems of ODEs 5.1 Systems of ODEs In a sense, Chapter 5 equals Chapter 2 “plus” Chapter 3, in the sense that Chapter 5 combines use of matrix theory and ordinary differential equation (ODE) methods. When we have more than one linear ODE, results from matrix theory turn out to be useful. Example 5.1 For the circuit shown in Figure 5.1, let v(t) be the voltage drop across the capacitor and I(t) be the loop current. The input V(t) is a given function. Assume, as usual, that L, R, and C are constants. Write down a system of ODEs in R2 that models this circuit. Method: The series RLC circuit shown in Figure 5.1 is analogous to the DC series RLC circuit discussed near the end of Section 3.3. The first ODE models the voltage drop across the capacitor being v(t) = 1 q(t),whereq(t) is the charge on the capacitor and C ˙ q˙(t) = I(t). The second ODE in the system is Kirchhoff’s voltage law, LI(t)+RI(t)+v(t) = V(t), after dividing through by L. The system is ⎧ ⎫ ⎨˙( ) = 1 ( ) ⎬ v t C I t ⎩ ⎭ . (5.1) ˙( ) = 1 ( ( ) − ( ) − ( )) I t L V t RI t v t More generally, consider a system of two ODEs in unknowns x1(t), x2(t): ⎧ ⎫ ˙ ⎨x1(t) = F1 t, x1(t), x2(t) ⎬ ⎩ ⎭ . (5.2) ˙ x2(t) = F2 t, x1(t), x2(t) A special case is ⎧ ⎫ ˙ ⎨x1(t) = a11(t)x1 + a12(t)x2 + f1(t)⎬ ⎩ ⎭ , (5.3) ˙ x2(t) = a21(t)x1 + a22(t)x2 + f2(t) which is called a linear system. In (5.3), we write x1 instead of x1(t) even though x1 is a function of t; we call this “suppressing the dependence on t” from the unknowns x1, x2. We will not suppress dependence on t in the coefficients aij(t) or the right-hand sides fi(t). 353 © 2014 by Taylor & Francis Group, LLC 354 Advanced Engineering Mathematics L I V(t) R υ(t) C FIGURE 5.1 RLC series circuit. The simplest such system is ⎧ ⎫ ˙ ⎨x1 = a11x1 + a12x2⎬ ⎩ ⎭ , (5.4) ˙ x2 = a21x1 + a22x2 where a11, a12, a21, a22 are constants. Chapter 5 will focus on linear constant coefficients homogeneous systems (LCCHS)and their linear nonhomogeneous analogues; in Chapter 18, we will look at nonlinear systems of ODEs, including how they relate to linear homogeneous systems of ODEs. More generally, we can study systems involving n unknowns, x1(t), x2(t), ..., xn(t). The system is in Rn if n is the number of unknown functions whose derivatives appear, assuming no derivatives higher than the first appear. So, (5.1) through (5.4) are all systems in R2. We can write system (5.2) compactly in vector form as x˙(t) = F t, x(t) , where we define ( ) ˙ ( ) ( ( ) ( )) ( ) x1 t ˙( ) d ( ) x1 t ( ) F1 t, x1 t , x2 t x t ( ) , x t x t ˙ ( ) ,andF t, x t ( ( ) ( )) . x2 t dt x2 t F2 t, x1 t , x2 t We can rewrite linear system (5.3) compactly in matrix–vector form as x˙(t) = A(t)x(t) + f(t), where we define a (t) a (t) f (t) A(t) 11 12 and f(t) 1 . a21(t) a22(t) f2(t) © 2014 by Taylor & Francis Group, LLC Linear Systems of ODEs 355 Here we extend the definition of multiplication of matrix times vector to functions of t: a (t) a (t) x (t) a (t)x (t) + a (t)x (t) A(t)x(t) = 11 12 1 11 1 12 2 . a21(t) a22(t) x2(t) a21(t)x1(t) + a22(t)x2(t) In particular, the system of ODEs (5.4) can be written compactly as x˙ = Ax, a a where the constant coefficient matrix is A = 11 12 . a21 a22 All of (5.1) through (5.4) can be generalized to systems in Rn, for example, x˙ = Ax can be short for ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ x˙ a ...a x ⎢ 1⎥ ⎢ 11 1n⎥ ⎢ 1⎥ ⎢ . ⎥ ⎢ .. .⎥ ⎢ . ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ . ⎥ = ⎢ ...⎥ ⎢ . ⎥ . ⎣ . ⎦ ⎣ ...⎦ ⎣ . ⎦ ˙ xn an1 ...ann xn Definition 5.1 A solution of an ODE system in Rn, x˙ = F(t, x), (5.5) is an n-vector of functions x(t) defined on an open interval I for which the derivative also exists on I and satisfies (5.5), that is, x˙(t) = F t, x(t) ,fort in I. Theorem 5.1 (Existence and uniqueness for solution of a linear system) If the matrix of functions A(t) and the vector of functions f (t) are continuous for t in an open interval I and t0 is inside I, then the initial value problem (IVP) for the linear system x˙ = A(t)x + f(t) x(t0)= x0 has exactly one solution on I. Example 5.2 ( ) ( ) For the circuit shown in Figure 5.2, let v1 t , v2 t be the voltage drops across the capaci- ( ) ( ) tors whose capacitances are C1, C2 and let I1 t , I2 t be the loop currents. Write down a R3 system of ODEs in that models this circuit, assuming L, R, C1,andC2 are, as usual, constants. Method: In the first loop, Kirchhoff’s voltage law gives ˙ ( ) + ( ) = ( ) LI1 t v1 t V t . (5.6) © 2014 by Taylor & Francis Group, LLC 356 Advanced Engineering Mathematics L R I I1 2 C v2(t) C2 V(t) v1(t) 1 FIGURE 5.2 RLC two-loop circuit. The input V(t) is a given function. In the second loop, Kirchhoff’s voltage law gives the algebraic equation ( ) + ( ) − ( ) = RI2 t v2 t v1 t 0, ( ) ( ) which we can solve for I2 in terms of v1 t , v2 t to get 1 I (t) = (v (t) − v (t)). (5.7) 2 R 1 2 In terms of the loop currents, the voltages across the capacitors satisfy ˙ ( ) = 1 ( − ) ˙ ( ) = 1 v1 t I1 I2 and v2 t I2. (5.8) C1 C2 Together, (5.6) through (5.8) give a linear system in R3, that is, a linear system of three ( ) ( ) ( ) ODEs in three unknowns, I1 t , v1 t ,andv2 t : ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ − 1 1 I (t) 0 L 0 I (t) V(t) ⎢ 1 ⎥ ⎢ ⎥ ⎢ 1 ⎥ ⎢ L ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ d ⎢ ⎥ ⎢ 1 1 1 ⎥ ⎢ ⎥ ⎢ ⎥ v (t) = ⎢ − ⎥ v (t) + . (5.9) ⎢ 1 ⎥ ⎢ C1 C1R C1R ⎥ ⎢ 1 ⎥ ⎢ 0 ⎥ dt ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ( ) 1 1 ( ) v2 t 0 − v2 t 0 C2R C2R Example 5.3 Rewrite as a linear system the ODE that modeled the spring–mass–damper system at the beginning of Section 3.3. Method: The spring–mass–damper system is modeled by ODE my¨ + by˙ + ky = 0. If we define the velocity of the mass by v = y˙, the physical situation is modeled by the system of two ODEs: ⎧ ⎫ ⎨ y˙ = v ⎬ ⎩ ⎭ . mv˙ = Forces =−bv − ky We can rewrite these ODEs in the matrix–vector form of a linear system in R2: ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ y 01y d ⎣ ⎦ = ⎣ ⎦ ⎣ ⎦ . (5.10) dt − k − b v m m v © 2014 by Taylor & Francis Group, LLC Linear Systems of ODEs 357 Example 5.4 Suppose an object has temperature T and it is in a medium whose temperature is M.In Section 3.1, we used Newton’s law of cooling, ˙ ( ) =− ( − ) T t kT T M , 1 where k is a constant dependent on the object’s material nature, in units of . T s Unlike Section 3.1, now let us assume that the temperature of the medium is affected by the object. Find a system of ODEs that models the whole situation. Method: Let us apply Newton’s law of cooling to the medium. We get ˙ ( ) =− ( − ) M t kM M T , where kM is a constant dependent on the medium’s material nature. So, the temperature of the medium affects the temperature of the object, which in turn affects the temperature of the medium: the temperatures of the object and the medium are intertwined. They satisfy the system of ODEs: − d T = kT kT T − . (5.11) dt M kM kM M We’ll assume that kT, kM are constants, which is reasonable as long as the temperatures are not changing too much and the materials are not changing their phases. 5.1.1 Systems of Second-Order Equations We saw that a second-order scalar ODE can be rewritten as a system of two first-order scalar ODEs. Newton’s second law of motion relating the acceleration of an object to the sum of the forces naturally leads to a second-order ODE. Similarly, if there are several objects, Newton’s law will apply to each of them, giving a system of second-order scalar ODEs. Just as for a single second-order scalar ODE, we can rewrite a system of m second-order scalar ODEs as a system of 2m first-order scalar ODEs. We will see that for certain systems of second-order scalar ODEs, it is simpler to leave them as first-order ODEs. Example 5.5 Describe the motion of the two objects, whose masses are m1 and m2, in the phys- ical system depicted in Figure 5.3. Assume that the system is in equilibrium when In equilibrium ℓ k1 k2 k3 ℓ +(x2 – x1) x x 1 2 x >0 x1 =0 x2 =0 k1 k2 k3 FIGURE 5.3 Two masses and three horizontal springs. © 2014 by Taylor & Francis Group, LLC 358 Advanced Engineering Mathematics = = x1 x2 0. As depicted in the picture, k1, k2, k3 are the spring constants of the three horizontal springs. Assume there are no damping forces. > Method: Assume x1 0 when the first object is to the right of its equilibrium position > > and similarly for x2 0.
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