Questions and Answers of A First Course in Differential Equations

(a) Consider the linear system X' = AX of three firstorder differential equations, where the coefficient matrix isand λ = 2 is known to be an eigenvalue of multiplicity two. Find two different
Verify that the vector X is a solution of the given system. 1 1' 0 X; X = -2 -1 X' = 6 -1 6 -13, 2.
In problem use a CAS or linear algebra software as an aid in finding the general solution of the given system. /0.9 2.1 3.2) X' = 0.7 6.5 4.2 X 1.1 1.7 3.4
In problem use variation of parameters to solve the given system. X' = X + -1 3, 1/
In problem use the method of Example 2 to compute eAt for the coefficient matrix. Use (1) to find the general solution of the given system. X' = (;) -2
Verify that is a solution of the linear systemfor arbitrary constants c1 and c2. By hand, draw a phase portrait of the system. x = )- C1 X C2.
In problem use a CAS or linear algebra software as an aid in finding the general solution of the given system. 1 -1.8 0 5.1 -1 3 X' = 1 -3 1 -3.1 4 -2.8 0 1.5 1 2. 2.
In problem use variation of parameters to solve the given system. 2 X + -1 3, X' = -31
In problem use the method of Example 2 to compute eAt for the coefficient matrix. Use (1) to find the general solution of the given system. '5 X' = -1
In problem the given vectors are solutions of a system X' = AX. Determine whether the vectors form a fundamental set on the interval (-∞, ∞). -2. X2 X, -1,
(a) Use computer software to obtain the phase portrait of the system in Problem 5. If possible, include arrowheads as in Figure 8.2.2. Also include four half-lines in your phase portrait.(b) Obtain
In problem use variation of parameters to solve the given system. -G -)x+ () 8 12) X' = ( 12
In problem use the method of Example 2 to compute eAt for the coefficient matrix. Use (1) to find the general solution of the given system. X' = -2 -2,
In problem the given vectors are solutions of a system X' = AX. Determine whether the vectors form a fundamental set on the interval (-∞, ∞). X, = (-)
Find phase portraits for the systems in Problems 2 and 4. For each system find any half-line trajectories and include these lines in your phase portrait.Problems 2 and 4. dx 2x + 2y dt dy = x + 3y dt
In problem use variation of parameters to solve the given system. x' = (; )x ) X + -1 te
In problem verify the foregoing result for the given matrix.Let P denote a matrix whose columns are eigenvectors K1, K2, . . . , Kn corresponding to distinct eigenvalues λ1, λ2, . . . , λn of an n
In problem the given vectors are solutions of a system X' = AX. Determine whether the vectors form a fundamental set on the interval (-∞, ∞). X, -2+ t 2 X, -2 2. 3' (2) X3 -6|+ t| 4 12 4. 4, 4.
Find the general solution of the given system. dx = 3x – y dt dy = 9x – 3y dt
In problem use variation of parameters to solve the given system. X' = )* - () 2e X + -1/ -2 3.
In problem verify the foregoing result for the given matrix.Let P denote a matrix whose columns are eigenvectors K1, K2, . . . , Kn corresponding to distinct eigenvalues λ1, λ2, . . . , λn of an n
In problem the given vectors are solutions of a system X' = AX. Determine whether the vectors form a fundamental set on the interval (-∞, ∞). 2 6 X2 = -2 X3 = 3 ,3t -13 -2
Find the general solution of the given system. dx -6x + 5y dt dy -5x + 4y dt
In problem use variation of parameters to solve the given system. 3 X'= X + -2 1- 1/ () -
Suppose A = PDP-1, where D is defined as in (9). Use (3) to show that eAt = PeDt P-1.
Verify that the vector Xp is a particular solution of the given system. dx = x + 4y + 2t - 7 dt dy 3x + 2y – 4t – 18; X, = dt 5. t + d.
Find the general solution of the given system. -1 3 X' = -3
In problem use variation of parameters to solve the given system. )x + sec t X' =
If D is defined as in (9), then find eDt.
Verify that the vector Xp is a particular solution of the given system. x = ; )x+ () x, - C -() X + %3D
Find the general solution of the given system. 12 -9 X' = 4
In problem use variation of parameters to solve the given system. X' = ; )x X + et 3
Find the general solution of the given system. X' = 3 x 2. mサー図
Verify that the vector X is a solution of the given system. dx 3x dt 4y dy = 4x – 7y; X = dt 5t (2)
In problem solve the given linear system. 2 8 2 X' = X + 0 4, 16t) %3D
In problem use (5) to find the general solution of the given system. X' = ( )x + (;) 1 0.
(a) The system of differential equations for the currents i2(t) and i3(t) in the electrical network shown in Figure 8.3.1 isUse the method of undetermined coefficients to solve the system if R1 = 2
Find the general solution of the given system. (1 0 1V X' = 0 1 0 X 101
Write the given system without the use of matrices. D) - ()- d (x sin t + 8 (2+ 1)
Solvesubject to 3. X + 3) -1 X' = 3 4
In problem use (5) to find the general solution of the given system. X' = X + let
In problem solve the given linear system. X' 1 12 2. 2.
Find the general solution of the given system. -1 1 1 2 X' = 1 X 0 3 -1/
Write the given system without the use of matrices. 1 -1 2\/x 3\ d y = -4 1 + 2Je 3 y+ dt -2 5 6/\z 12
In problem solve the given linear system. 1 -1 X' = 0 1 3 X 3 1, 4)
In problem use (5) to find the general solution of the given system. X' = 3 X + -1) 0 2/
In problem use the method of undetermined coefficients to solve the given system. /o 0 5 5 X' = 0 5 0X + 5 0 0/ - 10 40/
Find the general solution of the given system. dx = 2x – 7y dt dy 5x + 10y + 4z dt dz 5y + 2z dt
Write the given system without the use of matrices. 5 -9 /7 X' = |4 5 8' 1 1 e X + 2 -21 0 Je 10 -2 3, 3.
In problem solve the given linear system. -2 5 X'= 2 4,
In problem use (1) to find the general solution of the given system. o 0 0 X' = 3 0 0 x \5 1 0/
In problem use the method of undetermined coefficients to solve the given system. 1 1 1 X' = 0 2 3 X + e4t 2/ -1 0 0 5/
Find the general solution of the given system. dx = x + y - z dt dy 2y dt dz = y - z dt
Write the given system without the use of matrices. X' = x - (; 4 2 X + -1 3/ -1)
In problem solve the given linear system. 1 X' = -2
In problem use (1) to find the general solution of the given system. 1 X' = 1 1X -2 -2 -2,
In problem use the method of undetermined coefficients to solve the given system. -1 sin t X' X + -1 -2 cos t
Find the general solution of the given system. -96 X' = -3 -6 2
In problem write the linear system in matrix form. dx -3x + 4y + e-'sin 2t dt dy 5x + 9z + 4e'cos 2t dt dz = y + 6z – e-t dt
In problem solve the given linear system. dx -4x + 2y dt dy = 2x – 4y dt
In problem use (1) to find the general solution of the given system. X' = 1 0/
In problem use the method of undetermined coefficients to solve the given system. X' = X + 10/ %3D
Find the general solution of the given system. 10 X' = 8 -5 -12)
In problem write the linear system in matrix form. dx = x - y + z + t- 1 dt dy = 2x + y – z - 32 dt dz = x + y + z + t2 -t + 2 dt
In problem solve the given linear system. dx = 2x + y dt dy x- dt
In problem use (1) to find the general solution of the given system. X' = (, )x
In problem use the method of undetermined coefficients to solve the given system. X' = (; )» -4) X + 4t + 9e 4 -1 + e
Find the general solution of the given system. 2y 2y %3! 4 一山空山
In problem write the linear system in matrix form. = x - y x+ 2z -x + z | 小山山|咖山
Consider the linear system X' = AX of two differential equations, where A is a real coefficient matrix. What is the general solution of the system if it is known that λ1 = 1 + 2i is an eigenvalue
In problem use (3) to compute eAt. A = 3 0 5 1 0/
In problem use the method of undetermined coefficients to solve the given system. x = (; ;)x + (,) -2r %3D + + 5)
Find the general solution of the given system. dx -4x + 2y %3D dt dy * + 2y + 2y dt
In problem write the linear system in matrix form. dx -3x + 4y – 9 dt z %3D dy = 6x - y dt dz 10x + 4y + 3z dt
Consider the linear systemWithout attempting to solve the system, determine which one of the vectorsis an eigenvector of the coefficient matrix. What is the solution of the system corresponding to
In problem use (3) to compute eAt. 1 A = 1 1 1 -2 -2 -2,
In problem use (3) to compute eAt and e-At. A = 0/
In problem use the method of undetermined coefficients to solve the given system. dx = 5x + 9y + 2 dt dy -x + 1ly +6 dt
Find the general solution of the given system. dx = 2x + 2y dt dy = x + 3y dt
In problem write the linear system in matrix form. dx 4x - 7y dt dy 5x a dt
Fill in the blanks.The vector is solution of the initial-value problem for c1 = __________ and c2 = __________. -9t + C2 X = c 3.
In problem use (3) to compute eAt and e-At. 1 0) A 2)
In problem use the method of undetermined coefficients to solve the given system. dx 2x + 3y – 7 dt dy = -x - 2y + 5 dt
Find the general solution of the given system. dx = x + 2y dt dy = 4x + 3y dt
In problem write the linear system in matrix form. dx 3x – 5y dt dy = 4x + 8y dt
Fill in the blanks.The vector is a solution offor k = __________. X = k (5) 4.
(a) Assume that Theorem 7.3.1 holds when the symbol a is replaced by ki, where k is a real number and i2 = -1. Show that ℒ{tekti} can be used to deduce(b) Now use the Laplace transform to
Discuss how you would fix up each of the following functions so that Theorem 7.3.2 could be used directly to find the given Laplace transform. Check your answers using (16) of this section.(a) ℒ
Reread Example 4 in Section 3.1 on the cooling of a cake that is taken out of an oven.(a) Devise a mathematical model for the temperature of a cake while it is inside the oven based on the following
Abeam is embedded at its left end and simply supported at its right end. Find the deflection y(x) when the load is as given in Problem 77.
Find the deflection y(x) of a cantilever beam embedded at its left end and free at its right end when the load is as given in Example 10.
Solve Problem 77 when the load is given byProblem 77A cantilever beam is embedded at its left end and free at its right end. Use the Laplace transform to find the deflection y(x) when the load is
A cantilever beam is embedded at its left end and free at its right end. Use the Laplace transform to find the deflection y(x) when the load is given by Wo. 0
(a) Use the Laplace transform to find the charge q(t) on the capacitor in an RC-series circuit when q(0) = 0, R = 50 Ω, C = 0.01 f, and E(t) is as given in Figure 7.3.22.(b) Assume that E0 = 100 V.
(a) Use the Laplace transform to find the current i(t) in a single-loop LR-series circuit when i(0) = 0, L = 1 h, R = 10 Ω, and E(t) is as given in Figure 7.3.21.(b) Use a computer graphing program
In problem use the Laplace transform to find the charge q(t) on the capacitor in an RC-series circuit subject to the given conditions.q(0) = q0, R = 10 Ω, C = 0.1 f, E(t) given in Figure 7.3.20 E(1)
In problem use the Laplace transform to find the charge q(t) on the capacitor in an RC-series circuit subject to the given conditions.q(0) = q0, R = 10 Ω, C = 0.1 f, E(t) given in Figure 7.3.20 E(1)
Solve Problem 71 if the impressed force f (t) = sin t acts on the system for 0 ≤ t ≤ 2π and is then removed.q(0) = 0, R = 2.5Ω, C = 0.08 f, E(t) given in Figure 7.3.19. E() 5 + 3
Suppose a 32-pound weight stretches a spring 2 feet. If the weight is released from rest at the equilibrium position, find the equation of motion x(t) if an impressed force f(t) = 20t acts on the
The charge q(t) on a capacitor in an LC-series circuit is given byAppropriately modify the procedure of Problem 68 to find q(t). Graph your solution. d'q + q = 1 - 4U(t - ) + 6U(t – 3 ), dr q(0) =

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