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mathematics
first course differential equations

Differential Equations And Linear Algebra 4th Edition C. Edwards, David Penney, David Calvis - Solutions

Problems 43 through 47 pertain to the solution of differential equations with complex coefficients.Use the quadratic formula to solve the following equations. Note in each case that the roots are not complex conjugates.(a) x2 + ix + 2 = 0 (b) x2 - 2ix + 3 = 0
Apply Theorems 5 and 6 to find general solutions of the differential equations given in Problems 33 through 42. Primes denote derivatives with respect to x.y'' + 2y' - 15y = 0 THEOREM 5 Distinct Real Roots If the roots ₁ and 2 of the characteristic equation in (18) are real and
First verify by substitution that y1(x) = x-1/2 cos x is one solution (for x > 0) of Bessel's equation of order 1/2,x2y" + xy' + (x2 - 1/4)y = 0.Then derive by reduction of order the second solution y2(x) = x-1/2 sin x.
Problems 43 through 47 pertain to the solution of differential equations with complex coefficients.Find a general solution of y" - 2iy' + 3y = 0.
Use trigonometric identities to find general solutions of the equations in Problems 44 through 46.y'' + y' + y = sin x sin 3x
Each of Problems 43 through 48 gives a general solution y(x) of a homogeneous second-order differential equation ay'' + by' + cy = 0 with constant coefficients. Find such an equation.y(x) = c1 + c2e-10x
Each of Problems 43 through 48 gives a general solution y(x) of a homogeneous second-order differential equation ay'' + by' + cy = 0 with constant coefficients. Find such an equation.y(x) = c1e10x + c2e-10x
Use trigonometric identities to find general solutions of the equations in Problems 44 through 46.y'' + 9y = sin4 x
Problems 43 through 47 pertain to the solution of differential equations with complex coefficients.Find a general solution of y" - iy' + 6y = 0.
Solve the initial value problems in Problems 31 through 40.y" + y = cos x ; y(0) = 1, y'(0) = -1
Solve the initial value problems in Problems 31 through 40.y" - 2y' + 2y = x +1; y(0) = 3, y'(0) = 0
Each of Problems 43 through 48 gives a general solution y(x) of a homogeneous second-order differential equation ay'' + by' + cy = 0 with constant coefficients. Find such an equation.y(x) = c1e-10x + c2xe-10x
In Problems 33 through 36, one solution of the differential equation is given. Find the general solution.6y(4) + 5y(3) + 25y'' + 20y' + 4y = 0; y = cos 2x
In Problems 33 through 36, one solution of the differential equation is given. Find the general solution.3y(3) - 2y" + 12y' - 8y = 0; y = e2x/3
Given a mass m, a dashpot constant c, and a spring constant k, Theorem 2 of Section 5.1 implies that the equationhas a unique solution for t ≧ 0 satisfying given initial conditions x (0) = x0, x'(0) = v0. Thus the future motion of an ideal mass-spring-dashpot system is completely determined by
Given a mass m, a dashpot constant c, and a spring constant k, Theorem 2 of Section 5.1 implies that the equationhas a unique solution for t ≧ 0 satisfying given initial conditions x (0) = x0, x'(0) = v0. Thus the future motion of an ideal mass-spring-dashpot system is completely determined by
Apply Theorems 5 and 6 to find general solutions of the differential equations given in Problems 33 through 42. Primes denote derivatives with respect to x.2y'' - y' - y = 0 THEOREM 5 Distinct Real Roots If the roots ₁ and 2 of the characteristic equation in (18) are real and distinct, then y(x)
Find a function y(x) such that y(4) (x) = y(3)(x) for all x and y (0) = 18, y'(0) = 12, y" (0) = 13, and y(3) (0) = 7.
Given a mass m, a dashpot constant c, and a spring constant k, Theorem 2 of Section 5.1 implies that the equationhas a unique solution for t ≧ 0 satisfying given initial conditions x (0) = x0, x'(0) = v0. Thus the future motion of an ideal mass-spring-dashpot system is completely determined by
Apply Theorems 5 and 6 to find general solutions of the differential equations given in Problems 33 through 42. Primes denote derivatives with respect to x.4y'' + 8y' + 3y = 0 THEOREM 5 Distinct Real Roots If the roots ₁ and 2 of the characteristic equation in (18) are real and
Solve the initial value problems in Problems 31 through 40.y(3) - 2y" + y' = 1 + xex ; y(0) = y'(0) = 0, y" (0) = 1
In each of Problems 38 through 42, a differential equation and one solution y1 are given. Use the method of reduction of order as in Problem 37 to find a second linearly independent solution y2.x2y" + xy' - 9y = 0 (x > 0) ; y1(x) = x3
Solve the initial value problems in Problems 31 through 40.y" + 2y' + 2y = sin 3x ; y(0) = 2, y' (0) = 0
Apply Theorems 5 and 6 to find general solutions of the differential equations given in Problems 33 through 42. Primes denote derivatives with respect to x.4y'' + 4y' + y = 0 THEOREM 5 Distinct Real Roots If the roots ₁ and 2 of the characteristic equation in (18) are real and distinct, then y(x)
In each of Problems 38 through 42, a differential equation and one solution y1 are given. Use the method of reduction of order as in Problem 37 to find a second linearly independent solution y2.4y" - 4y' + y = 0 ; y1(x) = ex/2
In Problems 39 through 42, find a linear homogeneous constant coefficient equation with the given general solution.y(x) = (A + Bx + Cx2)e2x
Apply Theorems 5 and 6 to find general solutions of the differential equations given in Problems 33 through 42. Primes denote derivatives with respect to x.9y'' - 12y' + 4y = 0 THEOREM 5 Distinct Real Roots If the roots ₁ and 2 of the characteristic equation in (18) are real and
Solve the initial value problems in Problems 31 through 40.y(3) + y'' = x + e-x ; y(0) = 1, y'(0) = 0, y" (0) = 1
In each of Problems 38 through 42, a differential equation and one solution y1 are given. Use the method of reduction of order as in Problem 37 to find a second linearly independent solution y2.x2y" - x(x + 2)y' + (x + 2)y = 0 (x > 0) ; y1(x) = x
In Problems 39 through 42, find a linear homogeneous constant coefficient equation with the given general solution.y(x) = Ae2x + B cos 2x + C sin 2x
Apply Theorems 5 and 6 to find general solutions of the differential equations given in Problems 33 through 42. Primes denote derivatives with respect to x.6y'' - 7y' - 20y = 0 THEOREM 5 Distinct Real Roots If the roots ₁ and 2 of the characteristic equation in (18) are real and
Solve the initial value problems in Problems 31 through 40.y(4) - y = 5; y(0) = y'(0) = y" (0) = y(3) (0) = 0
Find a particular solution of the equation y(4)_y(3) - y" - y' - 2y = 8x5.
In each of Problems 38 through 42, a differential equation and one solution y1 are given. Use the method of reduction of order as in Problem 37 to find a second linearly independent solution y2.(x + 1)y" - (x + 2)y' + y = 0 (x > -1) ; y1(x) = ex
Apply Theorems 5 and 6 to find general solutions of the differential equations given in Problems 33 through 42. Primes denote derivatives with respect to x.35y'' - y' - 12y = 0 THEOREM 5 Distinct Real Roots If the roots ₁ and 2 of the characteristic equation in (18) are real and
Find the solution of the initial value problem consisting of the differential equation of Problem 41 and the initial conditions y (0) = y'(0) = y'" (0) = y(3) (0) = 0.
In each of Problems 38 through 42, a differential equation and one solution y1 are given. Use the method of reduction of order as in Problem 37 to find a second linearly independent solution y2.(1 - x2)y" + 2xy' - 2y = 0 (-1 < x < 1) ; y1(x) = x
First note that y1(x) = x is one solution of Legendre’s equation of order 1,Then use the method of reduction of order to derive the second solution (1-x2)y" - 2xy' + 2y = 0.
In Problems 39 through 42, find a linear homogeneous constant coefficient equation with the given general solution.y(x) = (A + Bx + Cx2) cos 2x + (D + Ex + Fx2) sin 2x
Problems 43 through 47 pertain to the solution of differential equations with complex coefficients.(a) Use Euler's formula to show that every complex number can be written in the form reiθ, where r ≧ 0 and -π < θ ≦ π.(b) Express the numbers 4, -2, 3i, 1+ i, and -1 + i√3 in the form
Apply Theorems 5 and 6 to find general solutions of the differential equations given in Problems 33 through 42. Primes denote derivatives with respect to x.2y'' + 3y' = 0 THEOREM 5 Distinct Real Roots If the roots ₁ and 2 of the characteristic equation in (18) are real and distinct, then y(x) =
In Problems 33 through 36, one solution of the differential equation is given. Find the general solution.9y(3) + 11y" + 4y' - 14y = 0; y = e-x sin x
Solve the initial value problems in Problems 31 through 40.y(4) - 4y" = x2 ; y(0) = y'(0) = 1, y" (0) = y(3) (0) = -1
In Problems 21 through 30, set up the appropriate form of a particular solution yp, but do not determine the values of the coefficients.y" + 3y' + 2y = x(e-x - e-2x)
According to Eq. (21), the amplitude of forced steady periodic oscillations for the system mx'' + cx' + kx = F0 cos ωt is given by(a) If c ≧ ccr/√2, where ccr = √4km, show that C steadily decreases as ω increases. (b) If c < ccr/√2, show that C attains a maximum value
A programmable calculator or a computer will be useful for Problems 11 through 16. In each problem find the exact solution of the given initial value problem. Then apply the Runge–Kutta method twice to approximate (to five decimal places) this solution on the given interval, first with step size
In Problems 1–8, first apply the formulas in (9) to find A-1. Then use A-1 (as in Example 5) to solve the system Ax = b.Example 5 = [ A = 15 16]. b = [3] 8 5 10
In Problems 1–4, two matrices A and B and two numbers c and d are given. Compute the matrix cA + dB. A = 2 -1 0-3 5 3.B 6 ‚B= -2 3 }],c=5,d=-3 7 15
This problem deals with a highly simplified model of a car of weight 3200 lb (mass m = 100 slugs in fps units). Assume that the suspension system acts like a single spring and its shock absorbers like a single dashpot, so that its vertical vibrations satisfy Eq. (4) with appropriate values of the
In Problems 13–22, the given vectors span a subspace V of the indicated Euclidean space. Find a basis for the orthogonal complement V⊥ of V.v1 = (1, 2, 3, 1, 3), v2 = (1, 3, 4, 3, 6), v3 = (2,2, 4, 3,5)
In Problems 1–4, determine whether the given vectors are mutually orthogonal.v1 = (3,-2, 3,-4), v2 = (6, 3, 4, 6), v3 = (17,-12, -21,3)
A 12-lb weight (mass m = 0.375 slugs in fps units) is attached both to a vertically suspended spring that it stretches 6 in. and to a dashpot that provides 3 lb of resistance for every foot per second of velocity.(a) If the weight is pulled down 1 ft below its static equilibrium position and then
Problems 24 through 34 deal with a mass–spring–dashpot system having position function x(t) satisfying Eq. (4). We write x0 = x(0) and v0 = x'(0) and recall thatThe system is critically damped, overdamped, or underdamped, as specified in each problem.Show in this case that p = c/(2m), w = k/m,
Solve the initial value problems given in Problems 21 through 26.2y(3) - 3y" - 2y' = 0; y(0) = 1, y'(0) = -1, y" (0) = 3
In Problems 21 through 24, a nonhomogeneous differential equation, a complementary solution yc , and a particular solution yp are given. Find a solution satisfying the given initial conditions.y" - 2y' + 2y = 2x; y(0) = 4, y'(0) = 8; yc = c1ex cos x + c2ex sinx ; yp = x + 1
Problems 24 through 34 deal with a mass–spring–dashpot system having position function x(t) satisfying Eq. (4). We write x0 = x(0) and v0 = x'(0) and recall thatThe system is critically damped, overdamped, or underdamped, as specified in each problem.Deduce from Problem 24 that the mass passes
Determine whether the pairs of functions in Problems 20 through 26 are linearly independent or linearly dependent on the real line.f(x) = sin2 x, g(x) = 1 - cos 2x
Determine whether the pairs of functions in Problems 20 through 26 are linearly independent or linearly dependent on the real line.f(x) = ex sinx, g(x) = ex cos x
Derive the steady periodic solution ofIn particular, show that it is what one would expect—the same as the formula in (20) with the same values of C and ω, except with sin(ωt - α) in place of cos(ωt - α). mx" + cx' + kx = Fosinot.
Solve the initial value problems given in Problems 21 through 26.3y(3) + 2y" = 0 ; y(0) = -1, y'(0) = 0, y" (0) = 1
Problems 24 through 34 deal with a mass–spring–dashpot system having position function x(t) satisfying Eq. (4). We write x0 = x(0) and v0 = x'(0) and recall thatThe system is critically damped, overdamped, or underdamped, as specified in each problem.Deduce from Problem 24 that x(t) has a local
Let Ly = y'' + py' + qy. Suppose that y1 and y2 are two functions such thatLy1 = f(x) and Ly2 = g(x).Show that their sum y = y1 + y2 satisfies the nonhomogeneous equation Ly = f(x) + g(x).
Determine whether the pairs of functions in Problems 20 through 26 are linearly independent or linearly dependent on the real line.f(x) = 2 cos x + 3 sinx, g(x) = 3 cos x - 2 sinx
Problems 24 through 34 deal with a mass–spring–dashpot system having position function x(t) satisfying Eq. (4). We write x0 = x(0) and v0 = x'(0) and recall thatThe system is critically damped, overdamped, or underdamped, as specified in each problem.Show in this case that p = c/(2m), w = k/m,
As indicated by the cart-with-flywheel example discussed in this section, an unbalanced rotating machine part typically results in a force having amplitude proportional to the square of the frequency ω.(a) Show that the amplitude of the steady periodic solution of the differential equation(with a
In Problems 21 through 30, set up the appropriate form of a particular solution yp, but do not determine the values of the coefficients.y" - 6y' + 13y = xe3x sin 2x
Solve the initial value problems given in Problems 21 through 26.y(3) + 10y" + 25y' = 0 ; y(0) = 3, y'(0) = 4, y" (0) = 5
In Problems 21 through 30, set up the appropriate form of a particular solution yp, but do not determine the values of the coefficients.y(4) + 5y'' + 4y = sin x + cos2x
Problems 24 through 34 deal with a mass–spring–dashpot system having position function x(t) satisfying Eq. (4). We write x0 = x(0) and v0 = x'(0) and recall thatThe system is critically damped, overdamped, or underdamped, as specified in each problem.If x0 = 0, deduce from Problem 27 that p =
Find general solutions of the equations in Problems 27 through 32. First find a small integral root of the characteristic equation by inspection; then factor by division.y(3) + 3y'' - 4y = 0
Find general solutions of the equations in Problems 27 through 32. First find a small integral root of the characteristic equation by inspection; then factor by division.2y(3) - y'' - 5y' - 2y = 0
In Problems 21 through 30, set up the appropriate form of a particular solution yp, but do not determine the values of the coefficients.y(4) + 9y" = (x2 + 1) sin 3x
Problems 24 through 34 deal with a mass–spring–dashpot system having position function x(t) satisfying Eq. (4). We write x0 = x(0) and v0 = x'(0) and recall thatThe system is critically damped, overdamped, or underdamped, as specified in each problem.Prove that in this case the mass can pass
Problems 24 through 34 deal with a mass–spring–dashpot system having position function x(t) satisfying Eq. (4). We write x0 = x(0) and v0 = x'(0) and recall thatThe system is critically damped, overdamped, or underdamped, as specified in each problem.If the damping constant c is small in
Find general solutions of the equations in Problems 27 through 32. First find a small integral root of the characteristic equation by inspection; then factor by division.y(3) + 27y = 0
Problems 24 through 34 deal with a mass–spring–dashpot system having position function x(t) satisfying Eq. (4). We write x0 = x(0) and v0 = x'(0) and recall thatThe system is critically damped, overdamped, or underdamped, as specified in each problem.(Underdamped) Show that in this case p =
In Problems 21 through 30, set up the appropriate form of a particular solution yp, but do not determine the values of the coefficients.(D - 1)3 (D2 - 4)y = xex + e2x +e-2x
Verify that y1 = x and y2 = x2 are linearly independent solutions on the entire real line of the equationx2y'' - 2xy' + 2y = 0,but that W(x , x2) vanishes at x = 0. Why do these observations not contradict part (b) of Theorem 3? THEOREM 3 Wronskians of Solutions Suppose that y₁, y2, ..., yn are n
Find general solutions of the equations in Problems 27 through 32. First find a small integral root of the characteristic equation by inspection; then factor by division.y(4) - y(3) + y" - 3y' - 6y = 0
In Problems 21 through 30, set up the appropriate form of a particular solution yp, but do not determine the values of the coefficients.y(4) - 2y" + y = x2 cos x
Problems 24 through 34 deal with a mass–spring–dashpot system having position function x(t) satisfying Eq. (4). We write x0 = x(0) and v0 = x'(0) and recall thatThe system is critically damped, overdamped, or underdamped, as specified in each problem.Show that the local maxima and minima
Find general solutions of the equations in Problems 27 through 32. First find a small integral root of the characteristic equation by inspection; then factor by division.y(3) + 3y'' + 4y' - 8y = 0
Solve the initial value problems in Problems 31 through 40.y" + 4y = 2x; y(0) = 1, y'(0) = 2
Problems 24 through 34 deal with a mass–spring–dashpot system having position function x(t) satisfying Eq. (4). We write x0 = x(0) and v0 = x'(0) and recall thatThe system is critically damped, overdamped, or underdamped, as specified in each problem.(Underdamped) Let x1 and x2 be two
Solve the initial value problems in Problems 31 through 40.y" + 3y' + 2y = ex ; y(0) = 0, y'(0) = 3
Find general solutions of the equations in Problems 27 through 32. First find a small integral root of the characteristic equation by inspection; then factor by division.y(4) + y(3) - 3y" - 5y' - 2y = 0
Apply Theorems 5 and 6 to find general solutions of the differential equations given in Problems 33 through 42. Primes denote derivatives with respect to x.y'' - 3y' + 2y = 0 THEOREM 5 Distinct Real Roots If the roots ₁ and 2 of the characteristic equation in (18) are real and distinct, then y(x)
In Problems 33 through 36, one solution of the differential equation is given. Find the general solution.y(3) + 3y" - 54y = 0; y = e3x
Solve the initial value problems in Problems 31 through 40.y" + 9y = sin 2x ; y(0) = 1, y'(0) = 0
Problems 24 through 34 deal with a mass–spring–dashpot system having position function x(t) satisfying Eq. (4). We write x0 = x(0) and v0 = x'(0) and recall thatThe system is critically damped, overdamped, or underdamped, as specified in each problem.A body weighing 100 lb (mass m = 3.125 slugs
In Problems 13 through 22, first verify that the given vectors are solutions of the given system. Then use the Wronskian to show that they are linearly independent. Finally, write the general solution of the system. [2] =*[2] =* *[5 F]-* =
Suppose that a crossbow bolt is shot straight upward with initial velocity 288 ft/s. If its deceleration due to air resistance is (0.04)v, then its height x(t) satisfies the initial value problemFind the maximum height that the bolt attains and the time required for it to reach this height. x"=-32
In Problems 1 through 16, apply the eigenvalue method of this section to find a general solution of the given system. If initial values are given, find also the corresponding particular solution. For each problem, use a computer system or graphing calculator to construct a direction field and
Use the method of Examples 5, 6, and 7 to find general solutions of the systems in Problems 11 through 20. If initial conditions are given, find the corresponding particular solution. For each problem, use a computer system or graphing calculator to construct a direction field and typical solution
In Problems 12 and 13, find the natural frequencies of the three-mass system of Fig. 7.5.1, using the given masses and spring constants. For each natural frequency ω, give the ratio a1:a2:a3 of amplitudes for a corresponding natural mode x1 = a1 cos ωt , x2 = a2 cos ωt , x3 = a3 cos ωt.m1 = m2
Find general solutions of the systems in Problems 1 through 22. In Problems 1 through 6, use a computer system or graphing calculator to construct a direction field and typical solution curves for the given system. 1 -5 -1 -5 1 -[- 0 0 4 52 -2 X
Use the method of Examples 5, 6, and 7 to find general solutions of the systems in Problems 11 through 20. If initial conditions are given, find the corresponding particular solution. For each problem, use a computer system or graphing calculator to construct a direction field and typical solution
In the system of Fig. 7.5.12, assume that m1 = 1, k1 = 50, k2 = 10, and F0 = 5 in mks units, and that ω = 10. Then find m2 so that in the resulting steady periodic oscillations, the mass m1 will remain at rest(!). Thus the effect of the second mass-and-spring pair will be to neutralize the effect
In Problems 13 through 22, first verify that the given vectors are solutions of the given system. Then use the Wronskian to show that they are linearly independent. Finally, write the general solution of the system. -3 x = [3²]x²x₁ = [30³] e3t X; X1 3e3t X2 X2 = 2e-2t e-21 [2
In Problems 1 through 16, apply the eigenvalue method of this section to find a general solution of the given system. If initial values are given, find also the corresponding particular solution. For each problem, use a computer system or graphing calculator to construct a direction field and

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