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Differential Equations and Linear Algebra 2nd edition Jerry Farlow, James E. Hall, Jean Marie McDill, Beverly H. West - Solutions

Hyperplane Basis Find a basis for the following hyperplane in R4: x + 3y -2z + 6w = 0?
Symmetric Matrices Find the dimension and exhibit a basis for the subspace of all symmetric matrices in M22?
Making New Bases from Old: In Problems 1-3, a vector space and a basis will be given. Construct a different basis from the given basis. Make them substantially different, so that at least two vectors are not multiples of vectors in the original basis?1. {iÌ…, jÌ…,
Basis for P2 Do the vectors [t2 + t + 1, t + 1, 1] form a basis for P2? If so, represent the polynomial 3t2 + 2t + 1 in terms of this basis?
True / False Questions: If false, give a counterexample or a brief explanation?(a) A solution set of a homogeneous system of linear algebraic equations, given byIs a subspace of R4. True or false? (b) The dimension of W is 4. True or false? (c) A basis for W is {[3, 0), [0, 2). [0, 1], [-1, 1]}.
Essay Questions3 Consider the homogeneous system of linear equationsAnd its solution set Write a paragraph or two describing the solution set W of the system of equations. Write a clear exposition that includes appropriate use of the words linear combination, span, subspace, vector space, linearly
Convergent Sequence Space4 Discuss the set V of all convergent sequences as a vector space, where the addition of "vectors" and multiplication by a scalar are defined by {an}+ {bn} = {an + bn} and c{an} = {can}. Describe the zero element and additive inverse elements for this vector space. Make a
Cosets in R3 Using the following definition, in Problems 1 and 2, find the W-cosets for the give vectors v̅ and given a graphical description of each? Cosets If W is a subspace of vector space that includes the origin, and; v̅ is a vector in Rn, then the W-coset of v̅, denoted v̅ + W, is the
More Cosets Asin Problems 85 and 86, describe the nature of a coset if the subspace W is a line through the origin. Illustrate for W = {[x1, x2, x3] | x1 = t, x2 = 3t, x3 = 2t} and v̅ = (1, - 2, 1).
Line in Function Space: Interpret the general solution of the differential equation y' + 2y = e-2t as a line in a suitable function space?
Mutual Orthogonality: Prove that nonzero mutually orthogonal vectors in a vector space V are automatically linearly independent?
The Undamped Oscillator: For Problems 1-3, find the simple harmonic motion described by the initial-value problem?1. + x = 0, x(0) = 1, (0) = 02. + x = 0, x(0) = 1, (0) = 13. + 9x = 0, x(0) = 1. (0) = 0
Alternate Forms for sinusoidal Oscillations: To derive the conversion formulas (7) and (8), use the identity cos(a - () = cos a cos ( + sin a sin ( From trigonometry to show that the family of sinusoidal oscillations A cos(W0t -() can be written in the form c1 cos w0t + c2 sin (0t. Where c1 = A
Single Wave Forms of Simple Harmonic Motion: Rewrite Problem 1-3 in the form A cos((0t - () using the conversion equations (7). 1. cos t + sin t 2. cos t - sin t 3. - cos t + sin t
Component Form of Simple Harmonic Motion: Rewrite Problems 1-3 in the form c1 cos (0t cos + c2 sin (0t using the conversation equations (8)? 1. 2 cos (2t - () 2. cos (t + ( / 3) 3. 3 cos (t - ( / 4)
Relating Graphs: For the oscillator DE + 0.25x = 0, Fig. 4.1.4 shown previously linked solution graphs and phase portrait. Parts (a), (b), (c), and (d) relate to that figure.(a) Mark on the phase portrait the starting points (where r = 0) for the trajectories shown. (b) Write explicit solutions for
Phase Portraits: For Problems 1-3, find (t) and then sketch the trajectory for the NP in the x phase plane, with arrows showing the direction of motion. (These are the equations of Problems 1-2 and 23-30.) Explain how and why your phase portraits differ from each other and from Fig 4.1.4.1. + x
Changing Frequencies Consider the undamped harmonic oscillator defined by + (20x = 0 with initial conditions x(0) = 4 and (0) = 0.(a) For (0 = 0.5, I, and 2, the corresponding trajectories are plotted in Fig. 4.1 .8(a) on the same tx-plane In Fig. 4.1.8(b) the corresponding trajectories an plotted
Detective Work Suppose that you have received the following two graphs without their equations. Show how you can infer the equations from graphical information.(a) The graph in Fig. 4.I.9(a) represents A cos(t - (). Determine A and 6 from the graph and write the equation of the curve in the form of
Pulling a Weight: An object of mass 2 kg, resting on a frictionless table, is attached to the wall by a spring as in Fig. 4.1.1. A force of 8 nt is applied to the mass, stretching the spring and moving the mass 0.5 m from its equilibrium position. The object is then released.(a) Find the resulting
Finding the Differential Equation A mass of 500 gm is suspended from the ceiling by a frictionless spring. The mass stretches the spring 50 cm in coming to its equilibrium position, where the mass acting down is balanced exactly by the restoring force acting up. The object is then pulled down an
Initial-Value Problems A 16-lb object is attached to the ceiling by a frictionless spring and stretches the spring 6 in. before coming to its equilibrium position. Formulate the initial-value problem describing the motion of the object under each of the following sets of conditions. Set x equal to
One More Weight: A 1 2-lb object attached to the ceiling by a frictionless spring stretches the spring 6 in. as it comes to its equilibrium. Find and solve the equation of motion if the object is initially pushed up 4 in. from its equilibrium and given an upward velocity of 2 ft/sec?
Comparing Harmonic Motions: An object on a table attached 10 spring and wall as in Fig. 4. 1. 1 is pulled to the right, stretching the spring, and released. The same object is then pulled twice as far and released. What is the relationship between the two simple harmonic motions? Will the period of
Testing Your Intuition Knowing (from Example I) what you now do about the damped harmonic oscillator equation m + b. + kx = 0 and the meaning of the parameters m. b. and k, consider Problems 1-2. How would you expect the solution of each equation to behave? Can you imagine a physical system being
LR-Circuit Consider the series LR-circuit shown in Fig. 4. 1. l 0, in which a constant input voltage V0 has been supplied until t = 0, when it is shut off.(a) Before carrying out the mathematical analysis, describe what you think will happen to the circuit. (b) For t > 0, use Kirchoff's voltage law
LC-Circuit: Consider the series LC-circuit shown in Fig. 4. 1. 11, in which, at t = 0, the current is 5 amps and there is no charge on the capacitor. Voltage V0 is turned off at r = 0.(a) Before carrying out the mathematical analysis, describe what you think will happen to the charge on the
A Pendulum Experiment A pendulum of length L is suspended from the ceiling so it can swing freely; 9 denotes the angular displacement, in radians, from the vertical, as shown in Fig. 4.1. 12. The motion is described by the pendulum equation,Determine the period for small oscillations by using the
Changing into Systems: For Problems 1-3, consider the second-order non-homogeneous DEs. Write them as a system of first order DEs as in (18). 1. 4- 2+ 3x = 17 - cos t 2. L+ R+ 1/C q = V(t) 3. 5+ 15+ 1/10 q = 5 cos3t
Circular Motion: A particle moves around the circle x2 + y2 = r2 with a constant angular velocity of (0 radians per unit time. Show that the projection of the particle on the x axis satisfies the equation + (0x = 0?
Another Harmonic Motion The mass-spring-pulley system shown in Fig. 4.1.13 satisfies the differential equationWhere x is the displacement from equilibrium of the object of mass m. In this equation, R and I are, respectively, the radius and moment of inertia of the pulley, and k is the spring
Motion of a Buoy A cylindrical buoy with diameter 1 8 in. floats in water with its axis vertical, as shown in Fig. 4.1.14. When depressed slightly and released, itsperiod of vibration is found to be 2.7 sec. Find the weight of the cylinder. Archimedes' Principle says that an object submerged in
Los Angeles to Tokyo It can be shown that the force on an object inside a spherical homogeneous mass is directed towards the center of the sphere with a magnitude proportional to the distance from the center of the sphere. Using this principle, a train starting at rest and traveling in a vacuum
Factoring Out Friction: The damped oscillator equation (2) can be solved by a change of variable that "factors out the damping" Specifically, let x(r) = e-(b/2m)t X(t).(a) Show that X(r) satisfies(b) Assuming that k - b2/4m > 0, solve equation for X(t); then show that the solution of equation (2) is
Graphing by Calculator: For the combinations of sine and cosine functions in Problems 1-3, do the following? (a) Use a graphing calculator or computer to sketch the graph of each function. (b) From your graphs, estimate the amplitude, period, and phase shift (/(0 of resulting oscillation. (c) Write
Real Characteristic Roots: Determine the general solutions for the differential equations in Problems 1-3? 1. y" = 0 2. y" - y' = 0 3. y" - 9y = 0
Initial Values Specified for Problems 1-3 solve the initial value problems? 1. y" - 25y = 0, y(0) = 1, y' (0) = 0 2. y" - y'- 2y = 0, y(0) = 1, y' (0) = 0 3. y" + 2y' + y, y(0) = 0, y' (0) = 0
Base and Solution Spaces: For each of the differential equation in problems 1-3, give a basis and a solution space in terms of the basis? 1. y" - 4y' = 0 2. y" - 10y' + 25 y = 0 3. 5y" - 10y' - l5y = 0
Other Bases: Use the Solution Space Theorem to show that the sets given in Problems 1 and 2 are each a basis for the DE? 1. y' - 4y = 0; {e2t, e-2t}, {cosh 2t, sinh 2t}, {e2t, cosh 2t} 2. y" = 0; {1, t}, {t + 1, t - 1}, {2t, 3t - 1}
The Wornskian Test: Use the Wronskian Test in Problems 1-3 to determine if the set of solutions is a basis for the given DE. 1. y(4) = 0, {t + 1, t - 1, t2 + t, t3} 2. y" - 10 y" - 15y' = 0. {te-5t, e5t, 2e5t - 1} 3. y(4) = 0, {t + 1, t2 + 2t, t2 - 2}
Sorting Graphs: For the DE + 5+ 6x = 0 of Example 1, Fig. 4.2.8 adds to Fig. 4.2.1 the linked solution graph for (t). Label the phase-plane trajectories from left to right as A, B, C. D, E. Then attach the same labels to the appropriate linked solutions x(t) and (t)?
Independent Solutions Show that if r1 and r2 are distinct real characteristic roots of equation (1), then the solutions e'' 1 and er2t are linearly independent?
Second Solution: Verify that if the discriminant of equation (1) as given by ( = b2 - 4ac is zero, so that b2 = 4ac and the characteristic root is r = - b/2a, then substituting y = v(t)e-(b/2a)r into ( I ) leads to the condition v"(t) = 0?
Independence Again: In the "repeated roots" case of equation (1), where ( = b2 - 4ac = 0 and r = - b/2a, show that the solutions e-(b/2)t and te-(b/2)t are linearly independent?
Repeated Roots, Long-Term Behavior: Show that in the "repeated roots" case of equation (1), the solution, which is given by x(t) = c1e-(b/2a)t + c2te-(b/2a)t , for b/2a > 0, tends toward zero as t becomes large. You may need l'Hpital's Rule: If, as x approachesa. both f(x) and g(x) approach zero,
Negative Roots: Verify that in the overdamped mass-spring system, for which ( = b2 - 4mk > 0, both characteristic roots are negative?
Circuits and Springs (a) What conditions on the resistance R, the capacitance C and the inductance L in equation (11) correspond to overdamping and critical damping in the mass-spring system? (b) Show that these conditions are directly analogous to b > (4mk for overdamping and b = (4mk for critical
A Test of Your Intuition We have two curves. The first starts at y(0) = 1 and its rate of increase equals its height: that is, it satisfies y' = y. The second curve also starts at y(0) = I with the same slope, and its second derivative, measuring upward curvature. equals its height; that is it
An Overdamped Spring The solution of the differential equation for an overdamped vibration has the form x(t) = c1et1 + c2er1t , with both c1 and c2 nonzero? (a) Show that x(t) is zero at most once. (b) Show that (1) is zero at most once.
A Critically Damped Spring The solution of the differential equation for a critically damped vibration has the from x(t) = (c1 + qt)ert with both c1 and c2 nonzero? (a) Show that x(t) is zero at most once. (b) Show that (t) is zero at most once.
Linking Graphs: For the sets of ty, ty', and yy' graphs in Problems 1-3, match the corresponding trajectories. They are numbered on the phase portrait, so you can use those same numbers to identify the curves in the component solution graphs. On each phase-plane trajectory, mark the point where t =
Damped Vibration: A small object of mass I slug rests on a frictionless table and is attaεhed, via a spring, to the wall. The damping constant is b = 2 lb sec/ft and the spring constant is k = I lb/ft. At time t = 0, the object is pulled 3 in. to the right and released. Show that the mass does not
Surge Functions The function x(t) = Ate-rt can be used to model events for which there is a surge and die-off; for example, the sales of a "hot" toy or the incidence of a highly infectious disease. This function can be obtained as the solution of a mass-spring system, m + b + kx = 0. Assume m = 1.
LRC-Circuit I A series LRC-circuit in a power grid has no input voltage, a resistor of IOI ohms, an inductor of 2 henries and a capacitor of 0.02 farads. Initially, the charge on the capacitor is 99 coulombs, and there is no current. (a) Determine the IVP for the charge across the capacitor. (b)
LRC-Circuit II: A series LRC-circuit with no input voltage has a resistor of 15 ohms, an inductor of 1 henry, and a capacitor of 0.02 farads. Initially, the charge on the capacitor is 5 coulombs, and there is no current. (a) Determine the IVP for the charge across the capacitor. (b) Solve the IVP
The Euler-Cauchy Equation A well-known linear second-order equation with variable coefficients is the Euler-Cauchy Equation3Where a, b, c ( IR. and a of; 0. Show by substituting y = tr that solutions of this form are obtained when r is a solution of the Euler-Cauchy characteristic equation Then
Euler-Cauchy Equations with Distinct Roots Obtain, for t > 0, the general solution of the Euler-Cauchy equations in Problems 1-3? 1. t2 y′′ + 2ty′ −12y = 0 2. 4t2y′′ + 8ty′ − 3y = 0 3. t2 y′′ + 4ty′ + 2y = 0
Repeated Euler-Cauchy Roots Verify that if the characteristic equation (15) for the Euler-Cauchy equation (14) has a repeated real root r, a second solution is given by tr In tr and that tr and tr In t are linearly independent?
Solutions for Repeated Euler-Cauchy Roots Obtain. For t > 0, the general solution of the Euler-Cauchy equations in Problems 1-3? 1. t2 y′′ + 5ty′ + 4y = 0 2. t2 y′′ − 3ty′ + 4y = 0 3. 9t2y″ + 3ty′ + y = 0
Computer: Phase-Plane Trajectories: Each of the functions in Problems 1-3 is the solution of a linear second-order differential equation with constant coefficients. In each case, do the following: (a) Determine the DE. (b) Calculate the derivative y' and the initial condition y(0), y'(0). (c) Plot
Reduction of Order4 For a solution y1 ofy" + p(x)y' + q(x)y = 0On interval I. such that y1 is not the zero function on I, use the following steps to find the conditions on a function v of x such thaty2 = uy1Is a solution to equation (16) that is linearly independent from y1 on I.(a) Determine y'2
Reduction of Order: Second Solution Use the steps o r the formula for y2 developed in Problem 75 to find a second linearly independent solution to the second-order differential equations of Problems 1-3 for which y1 is a known solution. Put the DE in standard form before using the formula. 1.
Classical Equations The equations in Problems 1-2 are some of the most famous differential equations in physics.5 Use d 'Alembert's reduction of order method described in Problem 75 along with the given solution y1 to find a second solution y2(t). Be prepared for integrals that you cannot
Lagrange's Adjoint Equation: The integrating factor method, which was an effective method for solving first order differentia1 equations, is not a viable approach for solving second-order equations. To see what happens even for the simplest equation, consider the differential equationy" + 3y' + 2y
Solutions in General: For Problems 1-10, determine tile general solution and give the basis B = {y1, .y2} for the solution space. 1. y" + 9y = 0 2. v" + v' + v = 0 3. y" - 4y' + 5y = 0
Initial-Value Problems: Solve the IVPs in Problems 1 - 3? 1. y" + 4y = 0, y(0) = 1 , y'(0) = - 1 2. y" - 4y' + 13y = 0, y(0) = 1. y'(0) = 0 3. y" + 2y' + 2y = 0, y(0) = I , y'(0) = 0
Working Backwanls Write the standard form (equation (16) with leading coefficient an = 11) of the nth order linear homogeneous differential equation with real coefficients whose roots are given in Problems 1-3? 1. y" + 4v = 0, y(0) = 1 , y'(0) = -1 2. y" - 4y' + 13y = 0, y(0) = 1 . y'(0) = 0 3. y"
Matching Problem: For Problems 1-3, determine which graph of the particular solution shown in Fig. 4.3.9 matches each deferential equation.1. y" - y' = 02. y" + y' = 03. y" + 3y' + 2y = 0(a)(b) (c) (d) (e) (f) g) (h) FIGURE4.3.9 Particular solutions that match the differential equations in Problems
Euler's Formula: You can use the following process to justify Euler's formulaei( = cos ( + i sin ((a) Write out explicitly the first dozen or so terms of the Maclaurin series (the Taylor expansion about the origin) given by(b) The series is valid for both real and complex numbers. Replace x by i(
Long-Term Behavior of Solutions: Suppose that r1 and r2 are the characteristic roots for ay" + by' + cy = 0, so the solution is y(t) = c1er1t. For Problems 1-2, discuss the long-term solution behaviors for the given r1. r2 combinations. Assume ( ( 0? 1. r1 < 0 , r2 < 0 2. r1 < 0, r2 = 0
Linear Independence: Verify that eat cos (t and eat sin (t are linearly independent on any interval?
Real Coefficients: Suppose the roots of the characteristic equation for (1) are complex conjugates a ± i(, which gives rise to the general solution y = k1e(a+i()t + k2e(a-i()t, where k1 and k2 are any constants (even complex). Show that in order for the solution y(t) to be real, k1 and k2 must be
Solving d" y /dt" = 0(a) Solve the equationby successive integration, getting d3y / dt3 = k3 and d2y / dt2 = k3t + k2 to obtain y(t). (b) Determine the characteristic equation for (18) and use its roots to find the general solution. Compare this solution with the solution you found in (a). (c)
Higher Order Des: Find the solutions for the following higher order equations. Remember that for each repetition of a root, a term with an additional factor of t must be included. If a factorization of the characteristic equation f(t) = 0 is not obvious, look for a small integer q that satisfies
Linking Graphs: For the sets of ty. ty', and yy' graphs in Problems 1 and 2, mach the corresponding trajectories. They are numbered on the phase portrait, so you can use those same numbers to identify the curves in the component solution graphs. On each phase-plane trajectory mark the point where r
Changing the Damping Consider the mass-spring system+ b + x = 0, x(0) = 4, (0) = 0.For damping coefficient b = 0, 0.5, 1, 2, 4, the corresponding solutions are plotted together in Fig. 4.3. 12(a) Their phase-plane trajectories are plotted in Fig. 4.3.l2(b) Make a trace of both graphs and label each
Changing the Spring Consider the mass-spring system+ + kx = 0, x(0) = 4, (0) = 0.(a) For spring constant k = 0.25, 0.5, 1, 2, 4, the corresponding solutions are plotted together in Fig. 4.3.13. (a) Their phase-plane trajectories are plotted in Fig. 4.3.13(b). Make a trace of both graphs and label
Changing the Mass A mass-spring system has a mass m attached in standard fashion with a damping factor b = 0 and a spring constant k = 1 6. (a) Discuss how the value of m affects the motion. (b) How would the frequency of the motion be affected if the mass were doubled? (c) Discuss how much damping
Oscillating Euler-Cauchy: Euler Cauchy equations were introduced in Sec. 4.2 Problem 60; Problems 1-2 consider Euler-Cauchy equations with non-real characteristic roots. The solutions then have the final form1. Verify the solution: Use the relation 2. Solve t2y" + 2ty' + y = 0
Third-Order Euler-Cauchy Use the substitution y = t' for t > 0 to obtain the characteristic equation for the following third-order Euler-Cauchy equation:
For the mapping defi11ed in each of Problems a to c, determine whether or not it is a linear tram-formation. a. T: R2 → R. T (x, y) = xy b. T: R2 → R2. T(x, y) = (x + y, 2y) c. T: R2 → R2. T(x, y) = (xy, 2y)
Integration Show that the integration operator I: C[a, b] †’ IR defined byis a linear transformation.
Show that the systems of linear differential equation given in problems a to b are linear transformations, where x = x(t) and y = y(t) 19 are linear ε C1 (I). a. T(x. y) = (x' - y, 2x + y') b. T(x. y) = (x + y', y -2x + y')
Determine whether or not the mappings in Problems a to c are linear transformations from R to R (a and b are real constants). a. T (x) = √x b. T (x) = ax + b c. T (x) = 1 / ax + b
For problems 1 to 3 Let T: R2 †’ R2 be the linear transformation given by T( ) = A, where1. verify that and explain why this means that the x-axis is mapped onto itself. 2. Verify that and explain why this means that the y-axis is mapped onto the line y = x/2. 3. Verify that and use this
Construct a matrix representation for the transformations in Problems a to c. and give a geometric interpretation of the mapping fromR2 to R2. Make sketches to illustrate your conclusions. a. T(x, y) = (x, -y) b. T(x, y) = (x, x) c. T(x, y) = (x, 0)
The composition T o S : U †’ W of two linear transformations T : V †’ W and S : U †’ V is defined byShow that the composition transformation is also a linear transformation.
For each linear transformation T: Rn †’ Rm in Problems 1 to 3, determine the standard matrix A such that1. T(x. y)= x + 2y 2. T(x. y)= (y. -x) 3. T(2, y) = (x + 2y,x - 2y)
For each linear transformation T: Rn †’ Rm given in Problems a to c. compute the image under T of . and find the vector(s) if any, that are mapped to .a. T(x, y) = (y, -x).b. T(x, y) = (x + y, x). c. T(x, y, z) = (x, y + z).
For r problems a to b, let T: R2 †’ R2 be defined byWherea. Determine the image under the map of the square having vertices (0, 0), (1, 0), (l , 1) , and (0, 1) . Calculate and compare the areas of the square and its image.b. Repeat problem 49 for the triangle with vertices (0, 0), (1, 1), and
Each matrix i n Problems 1 to 3 corresponds to one of the linear transformations in Problem 54. Match each matrix with the corresponding image from Fig. 5. 1.9.1. 2. 3.
Shear Transformation In Example 5, we looked at a shear transformation that produced a shear of one unit in the x-direction.(a) What linear transformation matrix would ·perform a shear of one unit in the y-direction on the r-shape in Fig. 5.1.6? Which image in Fig. 5.1.10 corresponds to this
In Example 6 we looked at a counterclockwise rotation about the origin. Write the matrix for a 30° clockwise rotation of the original r-shape. Graph the transformed r-shape.
Recall that linear transformations can be applied in succession by composition (de-fined in Problem 32). The corresponding process for linear transformation matrices is matrix multiplication. Consider a I -unit shear in the y-direction followed by a counter-clockwise rotation of 30°. Find the
(a) Reflect the r-shape (Fig. 5.1.6) about the x-axis and then about the y-axis.Find the transformation matrix M = RyRx. Where Rx is the matrix for reflection about the x-axis and Ry is the matrix for reflection about the y-axis, and sketch the transformed r-shape. (b) What counterclockwise
In the vector space Cˆž (a, b] of infinitely differentiable functions on the interval [a, b], consider the derivative transformation D and the definite integral transformation I defined by(a) Compute (DI) (f) = D(I(f)). (b) Compute (I D) (f) = 1(D(f)). (c) Do these transformations
The linear transformation T: R2 †’ R2 is defined byWherea. Determine the vectors in R2 that T maps to the zero vector in R4. b. Show that no vector in R2 is mapped to (1. I. I. I) in R4. c. Describe the subspace of le that is the image of T (that is, its range).
The linear transformation T: R3 †’ R2 is defined byWherea. Determine the vectors in R3 that maps to the zero vector in R2. b. Find the vectors in R3 that T maps to [I, I] in R2. c. Describe the image (range) of the transformation T.
Mappings from a vector space to the real numbers are sometimes called funetionals.3 Determine whether the transformations in Problems a to c are linear functionals from C[0. I] to R.a.b. c.
Verify that the mappings in problems 1 to 3 are linear transformations.1. L1: C1 †’ C, L1 (y) = y' + p(t) y2.(X and Y appropriate spaces) 3. (X the space of convergent real sequences)

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