Show that the Wronskians of two fundamental sets of solutions of the system (3) can

In Problems 1 through 6 you are given a homogeneous system of firstorder linear differential equations and two vector-valued functions, x(1) and x(2) . a. Show that the given functions are solutions of the given system of differential equations. b. Show that x = c1x(1) + c2x(2) is also a solution of the given system for any values of c1 and c2. c. Show that the given functions form a fundamental set of solutions of the given system. d. Find the solution of the given system that satisfies the initial condition x(0) = (1, 2) T . e. Find Wx(1) , x(2) (t). f. Show that the Wronskian, W = Wx(1) , x(2) , found in e is a solution of Abels equation: W = ( p11(t) + p22(t))W.x_ = _2 1 3 2 _ x; x(1) = _11_et , x(2) = _13 _et

In Problems 1 through 6 you are given a homogeneous system of firstorder linear differential equations and two vector-valued functions, x(1) and x(2) . a. Show that the given functions are solutions of the given system of differential equations. b. Show that x = c1x(1) + c2x(2) is also a solution of the given system for any values of c1 and c2. c. Show that the given functions form a fundamental set of solutions of the given system. d. Find the solution of the given system that satisfies the initial condition x(0) = (1, 2) T . e. Find Wx(1) , x(2) (t). f. Show that the Wronskian, W = Wx(1) , x(2) , found in e is a solution of Abels equation: W = ( p11(t) + p22(t))W.x_ = _1 1 4 2 _ x; x(1) = _ 1 4 _e3t , x(2) = _11 _e2t

In Problems 1 through 6 you are given a homogeneous system of firstorder linear differential equations and two vector-valued functions, x(1) and x(2) . a. Show that the given functions are solutions of the given system of differential equations. b. Show that x = c1x(1) + c2x(2) is also a solution of the given system for any values of c1 and c2. c. Show that the given functions form a fundamental set of solutions of the given system. d. Find the solution of the given system that satisfies the initial condition x(0) = (1, 2) T . e. Find Wx(1) , x(2) (t). f. Show that the Wronskian, W = Wx(1) , x(2) , found in e is a solution of Abels equation: W = ( p11(t) + p22(t))W.x_ = _2 5 1 2 _ x; x(1) = _ 5 cos t 2 cos t + sin t _, x(2) = _ 5 sin t 2 sin t cos t _

In Problems 1 through 6 you are given a homogeneous system of firstorder linear differential equations and two vector-valued functions, x(1) and x(2) . a. Show that the given functions are solutions of the given system of differential equations. b. Show that x = c1x(1) + c2x(2) is also a solution of the given system for any values of c1 and c2. c. Show that the given functions form a fundamental set of solutions of the given system. d. Find the solution of the given system that satisfies the initial condition x(0) = (1, 2) T . e. Find Wx(1) , x(2) (t). f. Show that the Wronskian, W = Wx(1) , x(2) , found in e is a solution of Abels equation: W = ( p11(t) + p22(t))W.x_ = _4 2 8 4 _ x; x(1) = _24 _, x(2) = _24 _t + _01 _

In Problems 1 through 6 you are given a homogeneous system of firstorder linear differential equations and two vector-valued functions, x(1) and x(2) . a. Show that the given functions are solutions of the given system of differential equations. b. Show that x = c1x(1) + c2x(2) is also a solution of the given system for any values of c1 and c2. c. Show that the given functions form a fundamental set of solutions of the given system. d. Find the solution of the given system that satisfies the initial condition x(0) = (1, 2) T . e. Find Wx(1) , x(2) (t). f. Show that the Wronskian, W = Wx(1) , x(2) , found in e is a solution of Abels equation: W = ( p11(t) + p22(t))W.tx_ = _2 1 3 2 _ x (t > 0); x(1) = _11 _t, x(2) = _13 _t1

In Problems 1 through 6 you are given a homogeneous system of firstorder linear differential equations and two vector-valued functions, x(1) and x(2) . a. Show that the given functions are solutions of the given system of differential equations. b. Show that x = c1x(1) + c2x(2) is also a solution of the given system for any values of c1 and c2. c. Show that the given functions form a fundamental set of solutions of the given system. d. Find the solution of the given system that satisfies the initial condition x(0) = (1, 2) T . e. Find Wx(1) , x(2) (t). f. Show that the Wronskian, W = Wx(1) , x(2) , found in e is a solution of Abels equation: W = ( p11(t) + p22(t))W.tx_ = _3 2 2 2 _ x (t > 0); x(1) = _12 _t1, x(2) = _21 _t2

Prove the generalization of Theorem 7.4.1, as expressed in the sentence that includes equation (8), for an arbitrary value of the integer k.

In this problem we outline a proof of Theorem 7.4.3 in the case n = 2. Let x(1) and x(2) be solutions of equation (3) for < t < , and let W be the Wronskian of x(1) and x(2) . a. Show that dW dt = ________ dx(1) 1 dt dx(2) 1 dt x(1) 2 x(2) 2 ________ + ________ x(1) 1 x(2) 1 dx(1) 2 dt dx(2) 2 dt ________ . b. Using equation (3), show that dW dt = ( p11 + p22)W. c. Find W(t) by solving the differential equation obtained in part b. Use this expression to obtain the conclusion stated in Theorem 7.4.3. d. Prove Theorem 7.4.3 for an arbitrary value of n by generalizing the procedure of parts a, b, and c.

Show that the Wronskians of two fundamental sets of solutions of the system (3) can differ at most by a multiplicative constant. Hint: Use equation (15). 1

If x1 = y and x2 = y_, then the second-order equation y__ + p(t) y_ + q(t) y = 0 (18) corresponds to the system x_ 1 = x2, x_ 2 = q(t) x1 p(t) x2. (19) Show that if x(1) and x(2) are a fundamental set of solutions of equations (19), and if y(1) and y(2) are a fundamental set of solutions of equation (18), then W[y(1) , y(2) ] = cW[x(1) , x(2) ], where c is a nonzero constant. Hint: y(1) (t) and y(2) (t) must be linear combinations of x11(t) and x12(t). 1

Show that the general solution of x_ = P(t)x + g(t) is the sum of any particular solution x( p) of this equation and the general solution x(c) of the corresponding homogeneous equation. 1

Consider the vectors x(1) (t) = _t1 _ and x(2) (t) = _t2 2t _. a. Compute the Wronskian of x(1) and x(2) . b. In what intervals are x(1) and x(2) linearly independent? c. What conclusion can be drawn about the coefficients in the system of homogeneous differential equations satisfied by x(1) and x(2) ? d. Find this system of equations and verify the conclusions of part c. 1

Consider the vectors x(1) (t) = _t2 2t _ and x(2) (t) = _et et _, and answer the same questions as in Problem 12. Problems 14 and 15 indicate an alternative derivation of Theorem 7.4.2. 1

Let x(1) , . . . , x(m) be solutions of x_ = P(t)x on the interval < t < . Assume that P is continuous, and let t0 be an arbitrary point in the given interval. Show that x(1) , . . . , x(m) are linearly dependent for < t < if (and only if) x(1) (t0), . . . , x(m) (t0) are linearly dependent. In other words, x(1) , . . . , x(m) are linearly dependent on the interval (, ) if they are linearly dependent at any point in it. Hint: There are constants c1, . . . , cm that satisfy c1x(1) (t0) + +cmx(m) (t0) = 0. Let z(t) = c1x(1) (t) + + cmx(m) (t), and use the uniqueness theorem to show that z(t) = 0 for each t in < t < . 1

Let x(1) , . . . , x(n) be linearly independent solutions of x_ = P(t)x, where P is continuous on < t < . a. Show that any solution x = z(t) can be written in the form z(t) = c1x(1) (t) + +cnx(n) (t) for suitable constants c1, . . . , cn. Hint: Use the result of Problem 10 of Section 7.3, and also Problem 14 above. b. Show that the expression for the solution z(t) in part a is unique; that is, if z(t) = k1x(1) (t) + + knx(n) (t), then k1 = c1, . . . , kn = cn. Hint: Show that (k1 c1)x(1) (t) + +(kn cn)x(n) (t) = 0 for each t in < t < , and use the linear independence of x(1) , . . . , x(n) .