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mathematics
first course differential equations

Questions and Answers of First Course Differential Equations

Find the convolution f(t) * g(t) in Problems 1 through 6.f(t) = g(t) = eat
Find the inverse Laplace transform f (t) of each function given in Problems 1 through 10. Then sketch the graph of f. F(s) = -πS e s²+1 2
Find the inverse Laplace transform f (t) of each function given in Problems 1 through 10. Then sketch the graph of f. F(s) = -$ se s²+7² 2
Apply the translation theorem to find the inverse Laplace transforms of the functions in Problems 5 through 10. F(s) = s-1 (s + 1)³
Apply the definition in (1) to find directly the Laplace transforms of the functions described (by formula or graph) in Problems 1 through 10.f(t) = sin2 t
Find the convolution f(t) * g(t) in Problems 1 through 6.f(t) = eat, g(t) = ebt (a ≠ b)
Use Laplace transforms to solve the initial value problems in Problems 1 through 16.x'' + 4x = cos t; x(0) = 0 = x' (0)
Use Laplace transforms to solve the initial value problems in Problems 1 through 16.x" + x = cos 3t; x (0) = 1, x'(0) = 0
Apply the convolution theorem to find the inverse Laplace transforms of the functions in Problems 7 through 14. F(s) = 1 s(s - 3)
Apply the translation theorem to find the inverse Laplace transforms of the functions in Problems 5 through 10. F(s) = I s² + 4s + 4 2
Find the inverse Laplace transform f (t) of each function given in Problems 1 through 10. Then sketch the graph of f. F(s) = 1-e-2лs s²+1
Use Laplace transforms to solve the initial value problems in Problems 1 through 16.x' = 2x + y, y' = 6x + 3y; x(0) = 1, y(0) = -2
Find the Laplace transforms of the functions given in Problems 11 through 22. f(t) = 2 if 0 ≤ t < 3; f(t) = 0 ift ≥ 3
Apply the convolution theorem to find the inverse Laplace transforms of the functions in Problems 7 through 14. F(s) = 5² (s² + 4)² 2
Use the transforms in Fig. 10.1.2 to find the Laplace transforms of the functions in Problems 11 through 22. A preliminary integration by parts may be necessary. f (t) = √t + 3t t f(t) t" (n
Use Laplace transforms to solve the initial value problems in Problems 1 through 16.x' = x + 2y, y' = x + e-t; x(0) = y(0) = 0
Use the transforms in Fig. 10.1.2 to find the Laplace transforms of the functions in Problems 11 through 22. A preliminary integration by parts may be necessary. f (t) = 3t5/2 - 4t3 t f(t) t" (n
Find the Laplace transforms of the functions given in Problems 11 through 22. f(t) = 1 if 1 ≤ t ≤ 4; f(t) = 0 ift < 1 or if t > 4
Apply the convolution theorem to find the inverse Laplace transforms of the functions in Problems 7 through 14. F(s) = 1 s(s² + 4s + 5)
Use Laplace transforms to solve the initial value problems in Problems 1 through 16.x' + 2y' + x = 0, x' - y' + y = 0; x (0) = 0, y(0) = 1
Find the Laplace transforms of the functions given in Problems 11 through 22. f(t) = sint if 0 ≤ t ≤ 2n; f(t) = 0 ift > 2n
Apply the convolution theorem to find the inverse Laplace transforms of the functions in Problems 7 through 14. F(s) = S (s - 3)(s² + 1)
Use the transforms in Fig. 10.1.2 to find the Laplace transforms of the functions in Problems 11 through 22. A preliminary integration by parts may be necessary. f (t) = t - 2e3t t f(t) t" (n
Use Laplace transforms to solve the initial value problems in Problems 1 through 16.x" + 2x + 4y = 0, y" + x + 2y = 0; x (0) = y(0) = 0, x' (0) = y'(0) = -1
Find the Laplace transforms of the functions given in Problems 11 through 22. f(t) = cos лt if 0 ≤ t ≤2; f(t) = 0 ift > 2
Apply the convolution theorem to find the inverse Laplace transforms of the functions in Problems 7 through 14. F(s) = S S4 + 5s² + 4
Use the transforms in Fig. 10.1.2 to find the Laplace transforms of the functions in Problems 11 through 22. A preliminary integration by parts may be necessary. f (t) = t3/2 - e-10t t f(t) t"
In Problems 13 through 20, a third-order homogeneous linear equation and three linearly independent solutions are given. Find a particular solution satisfying the given initial conditions. (3) +2y"
Suppose that the mass in a mass–spring–dashpot system with m = 10, c = 9, and k = 2 is set in motion with x(0) = 0 and x'(0) = 5.(a) Find the position function x(t) and show that its graph looks
In Problems 1 through 16, a homogeneous second-order linear differential equation, two functions y1 and y2, and a pair of initial conditions are given. First verify that y1 and y2 are solutions of
In Problems 1 through 20, find a particular solution yp of the given equation. In all these problems, primes denote derivatives with respect to x.y" + 2y' + 5y = ex sinx
In each of Problems 11 through 14, find and plot both the steady periodic solution xsp(t) = C cos(ωt - α) of the given differential equation and the actual solution x(t) = xsp(t) + Xtr(t) that
Find the general solutions of the differential equations in Problems 1 through 20.y(4) + 3y" - 4y = 0
In Problems 13 through 20, a third-order homogeneous linear equation and three linearly independent solutions are given. Find a particular solution satisfying the given initial conditions. y(3)-6y"
Suppose that the mass in a mass–spring–dashpot system with m = 25, c = 10, and k = 226 is set in motion with x(0) = 20 and x'(0) = 41.(a) Find the position function x(t) and show that its graph
In Problems 1 through 16, a homogeneous second-order linear differential equation, two functions y1 and y2, and a pair of initial conditions are given. First verify that y1 and y2 are solutions of
In Problems 1 through 20, find a particular solution yp of the given equation. In all these problems, primes denote derivatives with respect to x.y(4) - 2y" + y = xex
Each of Problems 15 through 18 gives the parameters for a forced mass-spring-dashpot system with equation mx" + cx' + kx = F0 cos ωt. Investigate the possibility of practical resonance of this
Find the general solutions of the differential equations in Problems 1 through 20.y(4) - 8y" + 16y = 0
In Problems 13 through 20, a third-order homogeneous linear equation and three linearly independent solutions are given. Find a particular solution satisfying the given initial conditions. y (3)-3y"
Find the Laplace transforms of the functions given in Problems 11 through 22. f(t) = sin лt if 2 ≤ t ≤ 3; f(t) = 0 if t < 2 or if t > 3
In Problems 15 through 22, apply either Theorem 2 or Theorem 3 to find the Laplace transform of f (t).f (t) = te2t cos 3t THEOREM 2 Differentiation of Transforms If f(t) is piecewise continuous for
Apply Theorem 2 to find the inverse Laplace transforms of the functions in Problems 17 through 24. F(s) = 1 s(s - 3)
Use the transforms in Fig. 10.1.2 to find the Laplace transforms of the functions in Problems 11 through 22. A preliminary integration by parts may be necessary. f (t) = cos2 2t t f(t) t" (n
In Problems 15 through 22, apply either Theorem 2 or Theorem 3 to find the Laplace transform of f (t).f (t) = te-t sin2 t THEOREM 2 Differentiation of Transforms If f(t) is piecewise continuous for
Find the Laplace transforms of the functions given in Problems 11 through 22. f(t) = cost if 3 ≤ t ≤ 5; f(t) = 0 ift < 3 or if t > 5
Apply Theorem 2 to find the inverse Laplace transforms of the functions in Problems 17 through 24. F(s) = = 3 s(s+ 5)
Use the transforms in Fig. 10.1.2 to find the Laplace transforms of the functions in Problems 11 through 22. A preliminary integration by parts may be necessary. f (t) = sin 3t cos 3t t f(t) t"
Find the Laplace transforms of the functions given in Problems 11 through 22. f(t) = 0 ift < 1; f(t) = t ift ≥ 1
Use Laplace transforms to solve the initial value problems in Problems 1 through 16.x' = x + z, y' = x + y, z' = -2x - z; x (0) = 1, y (0) = 0, z (0) = 0
Use the transforms in Fig. 10.1.2 to find the Laplace transforms of the functions in Problems 11 through 22. A preliminary integration by parts may be necessary. f (t) = sin 2t + cos 2t
In Problems 15 through 22, apply either Theorem 2 or Theorem 3 to find the Laplace transform of f (t).f (t) = t2 cos 2t THEOREM 2 Differentiation of Transforms If f(t) is piecewise continuous for
Find the Laplace transforms of the functions given in Problems 11 through 22. f(t) = sin 2t if ≤t ≤2n; f(t)= 0 ift < or if t > 2π
Use the transforms in Fig. 10.1.2 to find the Laplace transforms of the functions in Problems 11 through 22. A preliminary integration by parts may be necessary. f (t) = 1 + cosh 5t t f(t) t" (n
In Problems 15 through 22, apply either Theorem 2 or Theorem 3 to find the Laplace transform of f (t).f (t) = t sin 3t THEOREM 2 Differentiation of Transforms If f(t) is piecewise continuous for t≥
Find the Laplace transforms of the functions given in Problems 11 through 22. f(t) = sint if 0 ≤ t ≤ 3n; f(t) = 0 ift > 3r
Use Laplace transforms to solve the initial value problems in Problems 1 through 16.x" + x' + y' + 2x - y = 0, y" + x' + y' + 4x - 2y = 0; x (0) = y(0) = 1, x'(0) = y'(0) = 0
Solve the initial value problems in Problems 18 through 22. First make a substitution of the form t = x - a, then find a solution Σcntn of the transformed differential equation. State the interval
In Problems 19 through 30, express the general solution of the given differential equation in terms of Bessel functions.xy" + 3y' + xy = 0
Find two linearly independent Frobenius series solutions (for x > 0) of each of the differential equations in Problems 17 through 26.3xy'' + 2y' + 2y = 0
Solve the initial value problems in Problems 18 through 22. First make a substitution of the form t = x - a, then find a solution Σcntn of the transformed differential equation. State the interval
In Problems 19 through 22, first derive a recurrence relation giving cn for n ≧ 2 in terms of c0 or c1 (or both). Then apply the given initial conditions to find the values of c0 and c1. Next
In Problems 19 through 30, express the general solution of the given differential equation in terms of Bessel functions.xy" - y' + 36x3 y = 0
Find two linearly independent Frobenius series solutions (for x > 0) of each of the differential equations in Problems 17 through 26.2x2y'' + xy' - (1 + 2x2) y = 0
In Problems 19 through 22, first derive a recurrence relation giving cn for n ≧ 2 in terms of c0 or c1 (or both). Then apply the given initial conditions to find the values of c0 and c1. Next
Solve the initial value problems in Problems 18 through 22. First make a substitution of the form t = x - a, then find a solution Σcntn of the transformed differential equation. State the interval
In Problems 19 through 30, express the general solution of the given differential equation in terms of Bessel functions.x2y" - 5xy' + (8 + x) y = 0
Find two linearly independent Frobenius series solutions (for x > 0) of each of the differential equations in Problems 17 through 26.2x2y" + xy' - (3 - 2x2) y = 0
In Problems 19 through 22, first derive a recurrence relation giving cn for n ≧ 2 in terms of c0 or c1 (or both). Then apply the given initial conditions to find the values of c0 and c1. Next
In Problems 19 through 30, express the general solution of the given differential equation in terms of Bessel functions.36x2y" + 60xy' + (9x3 - 5) y = 0
Find two linearly independent Frobenius series solutions (for x > 0) of each of the differential equations in Problems 17 through 26.6x2y" + 7xy' - (x2 + 2)y = 0
In Problems 23 through 26, find a three-term recurrence relation for solutions of the form y = Σ cnxn. Then find the first three nonzero terms in each of two linearly independent solutions.y'' + (1
Show that the equationhas no power series solution of the form y = Σ cnxn. 2 x²y" + x²y + y = 0
In Problems 19 through 30, express the general solution of the given differential equation in terms of Bessel functions.16x2y" + 24xy' + (1 + 144x3) y = 0
Find two linearly independent Frobenius series solutions (for x > 0) of each of the differential equations in Problems 17 through 26.3x2y" + 2xy' + x2y = 0
In Problems 23 through 26, find a three-term recurrence relation for solutions of the form y = Σ cnxn. Then find the first three nonzero terms in each of two linearly independent solutions.(x2 - 1)
Establish the binomial series in (12) by means of the following steps.(a) Show that y = (1 + x)α satisfies the initial value problem (1 + x)y' = αy, y(0) = 1.(b) Show that the power series method
In Problems 19 through 30, express the general solution of the given differential equation in terms of Bessel functions.x2y" + 3xy' + (1 + x2)y = 0
Find two linearly independent Frobenius series solutions (for x > 0) of each of the differential equations in Problems 17 through 26.2xy" + (1 + x)y' + y = 0
In Problems 23 through 26, find a three-term recurrence relation for solutions of the form y = Σ cnxn. Then find the first three nonzero terms in each of two linearly independent solutions.y'' +
In Problems 19 through 30, express the general solution of the given differential equation in terms of Bessel functions.4x2y" - 12xy' + (15 + 16x) y = 0
Apply the extension of Theorem 1 in Eq. (22) to derive the Laplace transforms given in Problems 32 through 37.If f(t) is the sawtooth function whose graph is shown in Fig. 10.2.12, then L{f(t)}= =
Find two linearly independent Frobenius series solutions (for x > 0) of each of the differential equations in Problems 17 through 26.2xy" + (1 - 2x2)y' - 4xy = 0
Apply Theorem 1 as in Example 5 to derive the Laplace transforms in Problems 28 through 30. L{t cosh kt} = s²+k² (s²-k²)² ($2
Use Laplace transforms to solve the initial value problems in Problems 27 through 38.x" + 4x' + 8x = e-t; x(0) = x'(0) = 0
Use the transforms in Fig. 10.1.2 to find the inverse Laplace transforms of the functions in Problems 23 through 32. F(s) = 9+s 4-52
Use the transforms in Fig. 10.1.2 to find the inverse Laplace transforms of the functions in Problems 23 through 32. F(s) = 10s - 3 25-$2
Apply the results in Example 5 and Problem 28 to show thatProblem 28 - 1 (s²+k²) ² 1 38 2k-3 (sinkt - kt coskt).
In Problems 31 through 35, the values of mass m, spring constant k, dashpot resistance c, and force f (t) are given for a mass–spring–dashpot system with external forcing function. Solve the
Use the transforms in Fig. 10.1.2 to find the inverse Laplace transforms of the functions in Problems 23 through 32. F(s) = 25-1e-3s
In Problems 31 through 35, the values of mass m, spring constant k, dashpot resistance c, and force f (t) are given for a mass–spring–dashpot system with external forcing function. Solve the
Use Laplace transforms to solve the initial value problems in Problems 27 through 38.x(3) + x" - 6x' = 0; x (0) = 0, x'(0) = x" (0) = 1
Apply the extension of Theorem 1 in Eq. (22) to derive the Laplace transforms given in Problems 32 through 37. L{u(ta)} = s-le-as for a > 0.
In Problems 29 through 34, transform the given differential equation to find a nontrivial solution such that x(0) = 0.tx" + 2(t - 1)x' - 2x = 0
Use Laplace transforms to solve the initial value problems in Problems 27 through 38.x(4) - x = 0; x (0) = 1, x'(0) = x"(0) = x(3) (0) = 0
Derive the transform of f (t) = sin k t by the method used in the text to derive the formula in (16).
In Problems 31 through 35, the values of mass m, spring constant k, dashpot resistance c, and force f (t) are given for a mass–spring–dashpot system with external forcing function. Solve the
Apply the extension of Theorem 1 in Eq. (22) to derive the Laplace transforms given in Problems 32 through 37.If f(t) = 1 on the interval [a, b] (where 0 < a < b) and f(t) = 0 otherwise, then
In Problems 29 through 34, transform the given differential equation to find a nontrivial solution such that x(0) = 0.tx" - 2x' + tx = 0
Use Laplace transforms to solve the initial value problems in Problems 27 through 38.x(4) + x = 0; x (0) = x'(0) = x" (0) = 0, x(3) (0) = 1

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