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mathematics
first course differential equations

Questions and Answers of First Course Differential Equations

Suppose that m = 75 slugs (the car weighs 2400 lb), L1 = 7 ft, L2 = 3 ft (it's a rear-engine car), k1 = k2 = 2000 lb/ft, and I = 1000 ft-lb-s2. Then the equations in (40) take the form(a) Find the
Three 100-gal fermentation vats are connected as indicated in Fig. 7.1.11, and the mixtures in each tank are kept uniform by stirring. Denote by xi(t) the amount (in pounds) of alcohol in tank Ti at
In Problems 23 through 32 the eigenvalues of the coefficient matrix A are given. Find a general solution of the indicated system x' = Ax. Especially in Problems 29 through 32, use of a computer
Find the particular solution of the systemthat satisfies the initial conditions x1 (0) = 0, x2(0) = 0, x3 (0) = 17. dx1 dt dx2 dt dx3 dt 3x1 + x3, 9x1x2 + 2x3, = -9x1 + 4x2 - X3
In Problems 23 through 32 the eigenvalues of the coefficient matrix A are given. Find a general solution of the indicated system x' = Ax. Especially in Problems 29 through 32, use of a computer
In Problems 27 through 29, the system of Fig. 7.5.14 is taken as a model for an undamped car with the given parameters in fps units.(a) Find the two natural frequencies of oscillation (in hertz).(b)
A particle of mass m moves in the plane with coordinates (x (t), y(t)) under the influence of a force that is directed toward the origin and has magnitude k/(x2 + y2) - an inverse-square central
The phase portraits in Problems 17 through 28 correspond to linear systems of the form x' = Ax in which the matrix A has two linearly independent eigenvectors. Determine the nature of the eigenvalues
In Problems 23 through 32, find a particular solution of the indicated linear system that satisfies the given initial conditions.The system of Problem 17: x1 (0) = 8, x2 (0) = 0Data from in Problem
The amounts x1(t) and x2(t) of salt in the two brine tanks of Fig. 7.3.7 satisfy the differential equationswhere ki = r/Vi for i = 1, 2. In Problems 27 and 28 the volumes V1 and V2 are given. First
The amounts x1(t) and x2(t) of salt in the two brine tanks of Fig. 7.3.7 satisfy the differential equationswhere ki = r/Vi for i = 1, 2. In Problems 27 and 28 the volumes V1 and V2 are given. First
In Problems 23 through 32 the eigenvalues of the coefficient matrix A are given. Find a general solution of the indicated system x' = Ax. Especially in Problems 29 through 32, use of a computer
In Problems 27 through 29, the system of Fig. 7.5.14 is taken as a model for an undamped car with the given parameters in fps units.(a) Find the two natural frequencies of oscillation (in hertz).(b)
We can give a simpler description of the general solutionof the systemin Example 1 by introducing the oblique uv-coordinate system indicated in Fig. 7.4.17, in which the u- and v-axes are determined
Suppose that a projectile of mass m moves in a vertical plane in the atmosphere near the surface of the earth under the influence of two forces: a downward gravitational force FG of magnitude mg, and
The phase portraits in Problems 17 through 28 correspond to linear systems of the form x' = Ax in which the matrix A has two linearly independent eigenvectors. Determine the nature of the eigenvalues
In Problems 23 through 32 the eigenvalues of the coefficient matrix A are given. Find a general solution of the indicated system x' = Ax. Especially in Problems 29 through 32, use of a computer
The amounts x1(t) and x2(t) of salt in the two brine tanks of Fig. 7.3.8 satisfy the differential equationswhere ki = r/ Vi as usual. In Problems 29 and 30, solve for x1 (t) and x2 (t), assuming that
In Problems 23 through 32, find a particular solution of the indicated linear system that satisfies the given initial conditions.The system of Problem 18: x1 (0) = 0, x2 (0) = 0, x3 (0) = 4Data from
Suppose that a particle with mass m and electrical charge q moves in the xy-plane under the influence of the magnetic field B = Bk (thus a uniform field parallel to the z - axis), so the force on the
The amounts x1(t) and x2(t) of salt in the two brine tanks of Fig. 7.3.8 satisfy the differential equationswhere ki = r/ Vi as usual. In Problems 29 and 30, solve for x1 (t) and x2 (t), assuming that
In Problems 23 through 32 the eigenvalues of the coefficient matrix A are given. Find a general solution of the indicated system x' = Ax. Especially in Problems 29 through 32, use of a computer
In Problems 23 through 32, find a particular solution of the indicated linear system that satisfies the given initial conditions.The system of Problem 19: x1 (0) = 10, x2(0) = 12, x3 (0) = -1Data
Problems 35–37 show that all nontrivial solution curves of the system in Example 11 are ellipses rotated by the same angle as the trajectory in Fig. 7.4.11.The system in Example 11 can be rewritten
In Problems 23 through 32 the eigenvalues of the coefficient matrix A are given. Find a general solution of the indicated system x' = Ax. Especially in Problems 29 through 32, use of a computer
Problems 31 through 34 deal with the open three-tank system of Fig. 7.3.2. Fresh water flows into tank 1; mixed brine flows from tank 1 into tank 2, from tank 2 into tank 3, and out of tank 3; all at
The characteristic equation of the coefficient matrix A of the systemisTherefore, A has the repeated complex conjugate pair 3 ± 4i of eigenvalues. First show that the complex vectorsform a length 2
In Problems 23 through 32, find a particular solution of the indicated linear system that satisfies the given initial conditions.The system of Problem 21: x1 (0) = 1, x2 (0) = 2, x3 (0) = 3 x' =
In Problems 23 through 32 the eigenvalues of the coefficient matrix A are given. Find a general solution of the indicated system x' = Ax. Especially in Problems 29 through 32, use of a computer
In Problems 23 through 32 the eigenvalues of the coefficient matrix A are given. Find a general solution of the indicated system x' = Ax. Especially in Problems 29 through 32, use of a computer
(a) Show that the vector functionsare linearly independent on the real line.(b) Why does it follow from Theorem 2 that there is no continuous matrix P(t) such that x1 and x2 are both solutions of x'
Use the chain rule for vector-valued functions to verify the principle of time reversal.
Problems 31 through 34 deal with the open three-tank system of Fig. 7.3.2. Fresh water flows into tank 1; mixed brine flows from tank 1 into tank 2, from tank 2 into tank 3, and out of tank 3; all at
Verify Eq. (53) by substituting the expressions for x1(t) and x2 (t) from Eq. (51) into Eq. (52) and simplifying. x₁ (t) = 4 cos 10t - sin 10t, x2 (t) = 2 cos 10t + 2 sin 10t. (51)
In Problems 23 through 32, find a particular solution of the indicated linear system that satisfies the given initial conditions.The system of Problem 21: x1 (0) = 5, x2(0) = -7, x3 (0) = 11 x' =
Problems 31 through 34 deal with the open three-tank system of Fig. 7.3.2. Fresh water flows into tank 1; mixed brine flows from tank 1 into tank 2, from tank 2 into tank 3, and out of tank 3; all at
The characteristic equation of the coefficient matrix A of the systemisTherefore, A has the repeated complex conjugate pair 2 ± 3i of eigenvalues. First show that the complex vectorsform a length 2
In Problems 31–33 A represents a 2 x 2 matrix.(a) Show that if A has the repeated eigenvalue λ with two linearly independent associated eigenvectors, then every nonzero vector v is an
In Problems 31–33 A represents a 2 x 2 matrix.Use the definitions of eigenvalue and eigenvector (Section 7.3) to prove that if λ is an eigenvalue of A with associated eigenvector v, then -λ
In Problems 23 through 32, find a particular solution of the indicated linear system that satisfies the given initial conditions.The system of Problem 22: x1 (0) = x2(0) = x3 (0) = x4 (0) = 1Data
Suppose that one of the vector functionsis a constant multiple of the other on the open interval I. Show that their Wronskian W(t) = I[xij (t)]I must vanish identically on I. This proves part (a) of
Problems 31 through 34 deal with the open three-tank system of Fig. 7.3.2. Fresh water flows into tank 1; mixed brine flows from tank 1 into tank 2, from tank 2 into tank 3, and out of tank 3; all at
Find the position functions x1(t) and x2(t) of the railway cars of Fig. 7.5.2 if the physical parameters are given byand the initial conditions areHow far do the cars travel before stopping? m₁ =
In Problems 31–33 A represents a 2 x 2 matrix.Show that the system x' = Ax has constant solutions other than x(t) Ξ 0 if and only if there exists a (constant) vector x ≠ 0 with Ax = 0. (It is
Suppose that the vectors x1 (1) and x2 (1) of Problem 34 are solutions of the equation x' = P(t)x, where the 2 x 2 matrix P(t) is continuous on the open interval I. Show that if there exists a point
Problems 35 through 37 deal with the closed three-tank sys- tem of Fig. 7.3.5, which is described by the equations in (24). Mixed brine flows from tank 1 into tank 2, from tank 2 into tank 3, and
Problems 35–37 show that all nontrivial solution curves of the system in Example 11 are ellipses rotated by the same angle as the trajectory in Fig. 7.4.11.In analytic geometry it is shown that the
Problems 35 through 37 deal with the closed three-tank sys- tem of Fig. 7.3.5, which is described by the equations in (24). Mixed brine flows from tank 1 into tank 2, from tank 2 into tank 3, and
Problems 35–37 show that all nontrivial solution curves of the system in Example 11 are ellipses rotated by the same angle as the trajectory in Fig. 7.4.11.It can be further shown that Eq. (65)
Problems 35 through 37 deal with the closed three-tank sys- tem of Fig. 7.3.5, which is described by the equations in (24). Mixed brine flows from tank 1 into tank 2, from tank 2 into tank 3, and
Use the method of Examples 7 and 8 to find the Jordan normal form J of each coefficient matrix A given in Problems 23 through 32 (respectively). Example 7 In Example 3 we saw that the
Use the method of Examples 7 and 8 to find the Jordan normal form J of each coefficient matrix A given in Problems 23 through 32 (respectively). Example 7 In Example 3 we saw that the
Use the method of Examples 7 and 8 to find the Jordan normal form J of each coefficient matrix A given in Problems 23 through 32 (respectively). Example 7 In Example 3 we saw that the
Let A denote the 2 x 2 matrix(a) Show that the characteristic equation of A (Eq. (3), Section 6.1) is given by(b) Suppose that the eigenvalues of A are pure imaginary. Show that the trace T (A) = a
For each matrix A given in Problems 38 through 40, the zeros in the matrix make its characteristic polynomial easy to calculate. Find the general solution of x' = Ax. A = 1000 2 2 03 00 00 30 4 4
Let x1 (t), x2(t), ..., Xn (t) be vector functions whose ith components (for some fixed i) xi1 (1), xi2(t), ..., xin (t) are linearly independent real-valued functions. Conclude that the vector
For each matrix A given in Problems 38 through 40, the zeros in the matrix make its characteristic polynomial easy to calculate. Find the general solution of x' = Ax. A: -2 0 4 2 0
For each matrix A given in Problems 38 through 40, the zeros in the matrix make its characteristic polynomial easy to calculate. Find the general solution of x' = Ax. A = = 2 -21 0 0 0 0 -5
Use the eigenvalue/eigenvector method to confirm the solution in Eq. (61) of the initial value problem in Eq. (59).
Use projection matrices to find a fundamental matrix solution x(t) = eAt of each of the linear systems x' = Ax given in Problems 1 through 20Problem 3 x = 3x1 + 4x2, x = 3x1 + 2x2; x1(0) = x2(0) = 1
Use projection matrices to find a fundamental matrix solution x(t) = eAt of each of the linear systems x' = Ax given in Problems 1 through 20.Problem 4 x = 4x + x, x = 6x1-x2
Find a fundamental matrix of each of the systems in Problems 1 through 8, then apply Eq. (8) to find a solution satisfying the given initial conditions. x(t) = Φ(t)Φ(0) xo. = (8)
Apply the method of undetermined coefficients to find a particular solution of each of the systems in Problems 1 through 14. If initial conditions are given, find the particular solution that
Use projection matrices to find a fundamental matrix solution x(t) = eAt of each of the linear systems x' = Ax given in Problems 1 through 20.Problem 5 x = 6x17x2, x = x1-2x2
Apply the method of undetermined coefficients to find a particular solution of each of the systems in Problems 1 through 14. If initial conditions are given, find the particular solution that
Apply the method of undetermined coefficients to find a particular solution of each of the systems in Problems 1 through 14. If initial conditions are given, find the particular solution that
Use projection matrices to find a fundamental matrix solution x(t) = eAt of each of the linear systems x' = Ax given in Problems 1 through 20.Problem 7 x = 3x1 + 4x2, x = 6x15x2
Use projection matrices to find a fundamental matrix solution x(t) = eAt of each of the linear systems x' = Ax given in Problems 1 through 20.Problem 8 x = x - 5x, x = x1 - x x1 x2
Apply the method of undetermined coefficients to find a particular solution of each of the systems in Problems 1 through 14. If initial conditions are given, find the particular solution that
Use projection matrices to find a fundamental matrix solution x(t) = eAt of each of the linear systems x' = Ax given in Problems 1 through 20.Problem 9 x = = 2x15x2, x = 4x12x2; x1 (0) = 2, x2 (0) = 3
Apply the method of undetermined coefficients to find a particular solution of each of the systems in Problems 1 through 14. If initial conditions are given, find the particular solution that
Compute the matrix exponential eAt for each system x' = Ax given in Problems 9 through 20.x'1 = 5x1 - 4x2, x'2 = 2x1 - x2Data in Problem 9 through 20 9. x = 5x14x2, x = 2x1-x2 x = 4x14x2 10. x =
Use projection matrices to find a fundamental matrix solution x(t) = eAt of each of the linear systems x' = Ax given in Problems 1 through 20.Problem 12 x = x5x2, x = x + 3x
Compute the matrix exponential eAt for each system x' = Ax given in Problems 9 through 20.x'1 = 5x1 - 4x2, x'2 = 3x1 - 2x2Data in Problem 9 through 20 9. x = 5x14x2, x = 2x1-x2 x = 4x14x2 10. x =
Use projection matrices to find a fundamental matrix solution x(t) = eAt of each of the linear systems x' = Ax given in Problems 1 through 20.Problem 13 x = 5x19x2, x = 2x1-x2
Use projection matrices to find a fundamental matrix solution x(t) = eAt of each of the linear systems x' = Ax given in Problems 1 through 20.Problem 17 x = 4x + x2 + 4x3, x = x + 7x + x3, x3 = 4x1 +
In Problems 17 through 34, use the method of variation of parameters (and perhaps a computer algebra system) to solve the initial value problemIn each problem we provide the matrix exponential
Use projection matrices to find a fundamental matrix solution x(t) = eAt of each of the linear systems x' = Ax given in Problems 1 through 20.Problem 11 x = x - 2x2, x = 2x1 + x2; x (0) = 0, x2 (0) =
Compute the matrix exponential eAt for each system x' = Ax given in Problems 9 through 20.x'1 = 3x1 + x2, x'2 = x1 + 3x2Data in Problem 9 through 20 9. x = 5x14x2, x = 2x1-x2 x = 4x14x2 10. x =
Use projection matrices to find a fundamental matrix solution x(t) = eAt of each of the linear systems x' = Ax given in Problems 1 through 20.Problem 10 =-3x12x2, x = 9x1 + 3x2 x =
Compute the matrix exponential eAt for each system x' = Ax given in Problems 9 through 20.x'1 = 9x1 - 8x2, x'2 = 6x1 - 5x2Data in Problem 9 through 20 9. x = 5x14x2, x = 2x1-x2 x = 4x14x2 10. x =
Apply the method of undetermined coefficients to find a particular solution of each of the systems in Problems 1 through 14. If initial conditions are given, find the particular solution that
Apply the method of undetermined coefficients to find a particular solution of each of the systems in Problems 1 through 14. If initial conditions are given, find the particular solution that
Compute the matrix exponential eAt for each system x' = Ax given in Problems 9 through 20.x'1 = 5x1 - 3x2, x'2 = 2x1Data in Problem 9 through 20 9. x = 5x14x2, x = 2x1-x2 x = 4x14x2 10. x = 6x16x2,
Compute the matrix exponential eAt for each system x' = Ax given in Problems 9 through 20.x'1 = 6x1 - 6x2, x'2 = 4x1 - 4x2Data in Problem 9 through 20 9. x = 5x14x2, x = 2x1-x2 x = 4x14x2 10. x =
Find a fundamental matrix of each of the systems in Problems 1 through 8, then apply Eq. (8) to find a solution satisfying the given initial conditions. x(t) = Φ(t)Φ(0) xo. = (8)
Apply the method of undetermined coefficients to find a particular solution of each of the systems in Problems 1 through 14. If initial conditions are given, find the particular solution that
Use the method of Examples 7 and 8 to find the Jordan normal form J of each coefficient matrix A given in Problems 23 through 32 (respectively). Example 7 In Example 3 we saw that the
Find a fundamental matrix of each of the systems in Problems 1 through 8, then apply Eq. (8) to find a solution satisfying the given initial conditions. x(t) = Φ(t)Φ(0) xo. = (8)
Use projection matrices to find a fundamental matrix solution x(t) = eAt of each of the linear systems x' = Ax given in Problems 1 through 20.Problem 2 x = x2 2x1 + 3x2, x = 2x1 + x2
Use the method of Examples 7 and 8 to find the Jordan normal form J of each coefficient matrix A given in Problems 23 through 32 (respectively). Example 7 In Example 3 we saw that the
Use the method of Examples 7 and 8 to find the Jordan normal form J of each coefficient matrix A given in Problems 23 through 32 (respectively). Example 7 In Example 3 we saw that the
Use projection matrices to find a fundamental matrix solution x(t) = eAt of each of the linear systems x' = Ax given in Problems 1 through 20.Problem 1 x = x + 2x2, x = 2x + x
Use the method of Examples 7 and 8 to find the Jordan normal form J of each coefficient matrix A given in Problems 23 through 32 (respectively). Example 7 In Example 3 we saw that the
Apply the method of undetermined coefficients to find a particular solution of each of the systems in Problems 1 through 14. If initial conditions are given, find the particular solution that
In Problems 42 through 50, use a calculator or computer system to calculate the eigenvalues and eigenvectors in order to find a general solution of the linear system x' = Ax with the given
In Problems 42 through 50, use a calculator or computer system to calculate the eigenvalues and eigenvectors in order to find a general solution of the linear system x' = Ax with the given
In Problems 42 through 50, use a calculator or computer system to calculate the eigenvalues and eigenvectors in order to find a general solution of the linear system x' = Ax with the given
In Problems 42 through 50, use a calculator or computer system to calculate the eigenvalues and eigenvectors in order to find a general solution of the linear system x' = Ax with the given
Use the method of Examples 7 and 8 to find the Jordan normal form J of each coefficient matrix A given in Problems 23 through 32 (respectively). Example 7 In Example 3 we saw that the

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