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mathematics
first course differential equations

Questions and Answers of First Course Differential Equations

Apply the extension of Theorem 1 in Eq. (22) to derive the Laplace transforms given in Problems 32 through 37.If f(t) = (-1)[[t]] is the square-wave function whose graph is shown in Fig. 10.2.9,
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 unit on–off function whose graph is shown in Fig. 10.2.10, then
Use Laplace transforms to solve the initial value problems in Problems 27 through 38.x(4) + 13x" + 36x = 0; x (0) = x" (0) = 0, x' (0) = 2, x (3) (0) = -13
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
Derive the transform of f (t) = sinh k t by the method used in the text to derive the formula in (14).
In Problems 29 through 34, transform the given differential equation to find a nontrivial solution such that x(0) = 0.tx" + (4t - 2)x' + (13t - 4)x = 0
In Problems 36 and 37, a mass–spring–dashpot system with external force f (t) is described. Under the assumption that x(0) = x'(0) = 0, use the method of Example 7 to find the transient and
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 convolution theorem to show that 2et L 2-¹ { (s - D √✓s} = 2²/4 + √²/² S e-¹² du = e¹erf√i. π
Use Laplace transforms to solve the initial value problems in Problems 27 through 38.x(4) + 8x" + 16x = 0; x (0) = x' (0) = x" (0) = 0, x(3) (0) = 1
Apply the extension of Theorem 1 in Eq. (22) to derive the Laplace transforms given in Problems 32 through 37.If g(t) is the triangular wave function whose graph is shown in Fig. 10.2.11, then
In Problems 36 through 38, apply the convolution theorem to derive the indicated solution x(t) of the given differential equation with initial conditions x(0) = x'(0) = 0. x" + 4x = f(t); x (t)
Show that the function f(t) = sin(et2) is of exponential order as t → + ∞ but that its derivative is not.
In Problems 36 through 38, apply the convolution theorem to derive the indicated solution x(t) of the given differential equation with initial conditions x(0) = x'(0) = 0. x" + 2x + x = f(t);
Use Laplace transforms to solve the initial value problems in Problems 27 through 38.x" + 4x' + 13x = te-t ; x(0) = 0, x' (0) = 2
In Problems 36 and 37, a mass–spring–dashpot system with external force f (t) is described. Under the assumption that x(0) = x'(0) = 0, use the method of Example 7 to find the transient and
In Problems 36 through 38, apply the convolution theorem to derive the indicated solution x(t) of the given differential equation with initial conditions x(0) = x'(0) = 0. x" + 4x + 13x =
Use Laplace transforms to solve the initial value problems in Problems 27 through 38.x" + 6x' + 18x = cos 2t ; x (0) = 1, x'(0) = -1
The unit staircase function is defined as follows:(a) Sketch the graph of f to see why its name is appropriate.(b) Show thatfor all t ≧ 0.(c) Assume that the Laplace transform of the infinite
Suppose the function x(t) satisfies the initial value problem mx" + cx' + kx = F(t), x(a) = bo, x'(a) = b₁ for ta and x(t) = 0 for t < a. Then show that X(s) = L{x (t)} satisfies the equation m
Problems 39 and 40 illustrate two types of resonance in a mass–spring–dashpot system with given external force F(t) and with the initial conditions x(0) = x'(0) = 0.Suppose that m = 1, k = 9.04,
Given constants a and b, define h(t) for t ≧ 0 bySketch the graph of h and apply one of the preceding problems to show that h(t) = if n-1 ≤ t
Differentiate termwise the series for J0(x) to show directly that J'0(x) = -J1(x) (another analogy with the cosine and sine functions).
If x = a ≠ 0 is a singular point of a second-order linear differential equation, then the substitution t = x - a transforms it into a differential equation having t = 0 as a singular point. We then
In Problems 1 through 8, determine whether x = 0 is an ordinary point, a regular singular point, or an irregular singular point. If it is a regular singular point, find the exponents of the
Derive the recursion formula in Eq. (2) for Bessel’s equation. [(m + r)2 - p2]cm + Cm-2 = 0 (2)
Any integral of the form ∫ xm Jn (x) dx can be evaluated in terms of Bessel functions and the indefinite integral ∫ J0 (x) dx. The latter integral cannot be simplified further, but the function
Find general solutions in powers of x of the differential equations in Problems 1 through 15. State the recurrence relation and the guaranteed radius of convergence in each case.y'' + xy = 0 (an Airy
Any integral of the form ∫ xm Jn (x) dx can be evaluated in terms of Bessel functions and the indefinite integral ∫ J0 (x) dx. The latter integral cannot be simplified further, but the function
If x = a ≠ 0 is a singular point of a second-order linear differential equation, then the substitution t = x - a transforms it into a differential equation having t = 0 as a singular point. We then
Show (as in Example 3) that the power series method fails to yield a power series solution of the form y = Σ cnxn for the differential equations in Problems 15 through 18.xy' + y = 0 Example 3 Solve
If x = a ≠ 0 is a singular point of a second-order linear differential equation, then the substitution t = x - a transforms it into a differential equation having t = 0 as a singular point. We then
If x = a ≠ 0 is a singular point of a second-order linear differential equation, then the substitution t = x - a transforms it into a differential equation having t = 0 as a singular point. We then
Show (as in Example 3) that the power series method fails to yield a power series solution of the form y = Σ cnxn for the differential equations in Problems 15 through 18.2xy' = y Example 3 Solve
Any integral of the form ∫ xm Jn (x) dx can be evaluated in terms of Bessel functions and the indefinite integral ∫ J0 (x) dx. The latter integral cannot be simplified further, but the function
Use power series to solve the initial value problems in Problems 16 and 17.(1 + x2) y" + 2xy' - 2y = 0; y(0) = 0, y'(0) = 1
Show (as in Example 3) that the power series method fails to yield a power series solution of the form y = Σ cnxn for the differential equations in Problems 15 through 18.x2y' + y = 0 Example
Any integral of the form ∫ xm Jn (x) dx can be evaluated in terms of Bessel functions and the indefinite integral ∫ J0 (x) dx. The latter integral cannot be simplified further, but the function
Find two linearly independent Frobenius series solutions (for x > 0) of each of the differential equations in Problems 17 through 26.4xy'' + 2y' + y = 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
Show (as in Example 3) that the power series method fails to yield a power series solution of the form y = Σ cnxn for the differential equations in Problems 15 through 18.x3y' = 2y Example 3 Solve
Find two linearly independent Frobenius series solutions (for x > 0) of each of the differential equations in Problems 17 through 26.2xy'' + 3y' - y = 0
In Problems 19 through 30, express the general solution of the given differential equation in terms of Bessel functions.x2y'' - xy' + (1 + x2)y = 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
Find two linearly independent Frobenius series solutions (for x > 0) of each of the differential equations in Problems 17 through 26.2xy'' - y' - y = 0
Apply Theorem 2 to show that the Laplace transform of the square wave function of Fig. 10.5.12 is L{f(t)} = 1 s(1+e-as)'
Find the inverse transforms of the functions in Problems 23 through 28. F(s) = In s² +1 (s + 2)(s-3)
Use the factorizationto derive the inverse Laplace transforms listed in Problems 23 through 26. s4 + 4a4 = (s² - 2as +2a²) (s² + 2as + 2a²) ($2
Apply Theorem 2 to show that the Laplace transform of the sawtooth function f (t) of Fig. 10.5.13 is F(s) = 1 as2 e as s(1-e-as)*
Use the transforms in Fig. 10.1.2 to find the inverse Laplace transforms of the functions in Problems 23 through 32. F(s) = 1 S 2 85/2
Find the inverse transforms of the functions in Problems 23 through 28. F(s) = tan 1 3 s+2
Use the factorizationto derive the inverse Laplace transforms listed in Problems 23 through 26. s4 + 4a4 = (s² - 2as +2a²) (s² + 2as + 2a²) ($2
Use the transforms in Fig. 10.1.2 to find the inverse Laplace transforms of the functions in Problems 23 through 32. F(s) = 1 S+5
Find the inverse transforms of the functions in Problems 23 through 28. F (s) = ln (1 + -3/2)
Let g(t) be the staircase function of Fig. 10.5.14. Show that g(t) = (t/a) - f(t), where f is the sawtooth function of Fig. 10.5.14, and hence deduce that L{g(t)} e -as s(1-e-as)
Use Laplace transforms to solve the initial value problems in Problems 27 through 38.x" + 6x' + 25x = 0; x (0) = 2, x' (0) = 3
Use the transforms in Fig. 10.1.2 to find the inverse Laplace transforms of the functions in Problems 23 through 32. F(s) = 3 S-4
Find the inverse transforms of the functions in Problems 23 through 28. F(s) = S (s² + 1)3 2
Apply Theorem 1 as in Example 5 to derive the Laplace transforms in Problems 28 through 30. L{t cos kt} = s²-k² (s²+k²)²
Suppose that f(t) is a periodic function of period 2a with f(t) = t if 0 ≦ t < a and f (t) = 0 if a ≦ t < 2a. Find ℒ {f (t)}.
Use the transforms in Fig. 10.1.2 to find the inverse Laplace transforms of the functions in Problems 23 through 32. F(s) = 3s +1 s² +4
Use Laplace transforms to solve the initial value problems in Problems 27 through 38.x" - 6x' + 8x = 2; x (0) = x'(0) = 0
In Problems 29 through 34, transform the given differential equation to find a nontrivial solution such that x(0) = 0.tx'' + (t - 2)x' + x = 0
Apply Theorem 1 as in Example 5 to derive the Laplace transformsin Problems 28 through 30. £{t sinhkt}= 2ks (s²-k²)2
Suppose that f (t) is the half-wave rectification of sin kt , shown in Fig. 10.5.15. Show that L{f(t)} = k (s2+k²)(1-e-ns/k)*
Use Laplace transforms to solve the initial value problems in Problems 27 through 38.x" - 4x = 3t; 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) = 5-3s s² +9
Let g(t) = u(t - π/k) f (t - π/k), where f(t) is the function of Problem 29 and k > 0. Note that h(t) = f(t) + g(t) is the full-wave rectification of sin kt shown in Fig. 10.5.16. Hence deduce
In Problems 29 through 34, transform the given differential equation to find a nontrivial solution such that x(0) = 0.tx'' + (3t - 1)x' + 3x = 0
In Problems 19–22, reduce the given system to echelon form to find a single solution vector u such that the solution space is the set of all scalar multiples of u. 3x4 = 0 4x4 = 0 - 3x15x2x3 - 5x4
In Problems 19–22, reduce the given system to echelon form to find a single solution vector u such that the solution space is the set of all scalar multiples of u. x1 + 5x2 + x3 - 8x4 = 0 2x1 +
In Problems 15–18, apply the method of Example 5 to find two solution vectors u and v such that the solution space is the set of all linear combinations of the form su + tv. x14x2 + x3 4x4 = 0 x1 +
In Problems 1–14, a subset W of some n-space Rn is defined by means of a given condition imposed on the typical vector (x1 , x2 , .... ,xn). Apply Theorem 1 to determine whether or not W
In Problems 1–14, a subset W of some n-space Rn is defined by means of a given condition imposed on the typical vector (x1 , x2 , .... ,xn). Apply Theorem 1 to determine whether or not W
In Problems 1–14, a subset W of some n-space Rn is defined by means of a given condition imposed on the typical vector (x1 , x2 , .... ,xn). Apply Theorem 1 to determine whether or not W
In Problems 1–14, a subset W of some n-space Rn is defined by means of a given condition imposed on the typical vector (x1 , x2 , .... ,xn). Apply Theorem 1 to determine whether or not W
In Problems 1–14, a subset W of some n-space Rn is defined by means of a given condition imposed on the typical vector (x1 , x2 , .... ,xn). Apply Theorem 1 to determine whether or not W
In Problems 1–14, a subset W of some n-space Rn is defined by means of a given condition imposed on the typical vector (x1 , x2 , .... , xn). Apply Theorem 1 to determine whether or not W
In Problems 1–14, a subset W of some n-space Rn is defined by means of a given condition imposed on the typical vector (x1 , x2 , .... ,xn). Apply Theorem 1 to determine whether or not W
In Problems 1–14, a subset W of some n-space Rn is defined by means of a given condition imposed on the typical vector (x1, x2, ....,xn). Apply Theorem 1 to determine whether or not W is a subspace
In Problems 1–14, a subset W of some n-space Rn is defined by means of a given condition imposed on the typical vector (x1 , x2 , .... ,xn). Apply Theorem 1 to determine whether or not W
In Problems 1–14, a subset W of some n-space Rn is defined by means of a given condition imposed on the typical vector (x1, x2, ....,xn). Apply Theorem 1 to determine whether or not W is a subspace
In Problems 1–14, a subset W of some n-space Rn is defined by means of a given condition imposed on the typical vector (x1 , x2 , .... ,xn). Apply Theorem 1 to determine whether or not W
In Problems 1–14, a subset W of some n-space Rn is defined by means of a given condition imposed on the typical vector (x1 , x2 , .... ,xn). Apply Theorem 1 to determine whether or not W
In Problems 1–14, a subset W of some n-space Rn is defined by means of a given condition imposed on the typical vector (x1 , x2 , ....,xn). Apply Theorem 1 to determine whether or not W
In Problems 15–26, find a basis for the solution space of the given homogeneous linear system. x13x2 x1 +4x2+ 11x3 x1 + 3x2 + 8x3 10x3 + 5x4 = 0 2x4 = 0 x4 = 0
In Problems 15–26, find a basis for the solution space of the given homogeneous linear system. x14x2 3x3 - - + x3 + 7x4 = 0 2x1x₂ 7x4 = 0 X1 + 2x2 + 3x3 + 11x4 = 0 =
In Problems 15–26, find a basis for the solution space of the given homogeneous linear system. x1 + 5x2 + 13x3 + 14x4 = 0 2x1 + 5x2 + 11x3 + 12x4 = = 0 2x17x2 + 17x3 + 19x4 = 0
In Problems 15–26, find a basis for the solution space of the given homogeneous linear system. x1 + 3x2 - 4x3 X1 + 2x3 + 2x17x210x3 8x4 + 6x5 = 0 X4+ 3x50 19x4 + 13x5 = 0
In Problems 15–26, find a basis for the solution space of the given homogeneous linear system. x1 + 2x2 + 7x3 9x4 + 31x5 = 0 2x1 + 4x2 + 7x3 - 11x4 + 34x5: = 0 3x1 + 6x2 + 5x3 - 11x4 + 29x5 = 0
In Problems 12–14, find a basis for the indicated subspace of R4.The set of all vectors of the form (a , b , c , d) for which a + 2b = c + 3d = 0.
In Problems 12–14, find a basis for the indicated subspace of R4.The set of all vectors of the form (a , b , c , d) such that a = 3c and b = 4d.
In Problems 12–14, find a basis for the indicated subspace of R4.The set of all vectors of the form (a , b , c , d) for which a = b + c + d.
In Problems 9–11, find a basis for the indicated subspace of R3.The plane with equation x - 2y + 5z = 0.
In Problems 1–8, determine whether or not the given vectors in Rn form a basis for Rn.v1 = (2 , 0 , 0 , 0) , v2 = (0 , 3 , 0 , 0) , v3 = (0 , 0 , 7, 6) , v4 = (0 , 0 , 4 , 5)
In Problems 1–8, determine whether or not the given vectors in Rn form a basis for Rn.v1 = (0 , 0 , 1) , v2 = (7 , 4 , 11) , v3 = (5 , 3 , 13)
In Problems 1–8, determine whether or not the given vectors in Rn form a basis for Rn.v1 = (0 , 0 , 1) , v2 = ( 0, 1 , 2) , v3 = (1 , 2 , 3)
In Problems 1–8, determine whether or not the given vectors in Rn form a basis for Rn.v1 = (0 , 7 , -3), v2 = (0 , 5 , 4), v3 = (0 , 5 , 10)
In Problems 1–8, determine whether or not the given vectors in Rn form a basis for Rn.v1 = (3 . -7 , 5 , 2) , v2 = (1 , -1 , 3 , 4 ) , v3 = (7 , 11 , 3 , 13)

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