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mathematics
first course differential equations

Questions and Answers of First Course Differential Equations

In Problems 1 through 10, find a power series solution of the given differential equation. Determine the radius of convergence of the resulting series, and use the series in Eqs. (5) through (12) to
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
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.(2 - x2)y'' - xy' +
In Problems 1 through 10, find a power series solution of the given differential equation. Determine the radius of convergence of the resulting series, and use the series in Eqs. (5) through (12) to
Deduce the identities in Eqs. (24) and (25) from those in Eqs. (22) and (23). dx [x² Jp(x)] = xP Jp-1(x). (22)
Use the relation Γ(x + 1) = xΓ(x) to deduce from Eqs. (13) and (14) that if p is not a negative integer, thenThis form is more convenient for the computation of Jp(x) because only the single value
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
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.(x2 - 1)y'' + 8xy' +
In Problems 1 through 10, find a power series solution of the given differential equation. Determine the radius of convergence of the resulting series, and use the series in Eqs. (5) through (12) to
Use the series of Problem 9 to findifProblem 9Use the relation Γ(x + 1) = xΓ(x) to deduce from Eqs. (13) and (14) that if p is not a negative integer, thenThis form is more convenient for the
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 10, find a power series solution of the given differential equation. Determine the radius of convergence of the resulting series, and use the series in Eqs. (5) through (12) to
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.3y'' + xy' - 4y = 0
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
In Problems 11 through 14, use the method of Example 4 to find two linearly independent power series solutions of the given differential equation. Determine the radius of convergence of each series,
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.5y'' - 2xy' + 10y = 0
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
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
In Problems 11 through 14, use the method of Example 4 to find two linearly independent power series solutions of the given differential equation. Determine the radius of convergence of each series,
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'' - x2y' - 3xy = 0
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
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
In Problems 11 through 14, use the method of Example 4 to find two linearly independent power series solutions of the given differential equation. Determine the radius of convergence of each series,
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'' + x2y' + 2xy = 0
In Problems 1 through 8, apply the successive approximation formula to compute yn (x) for n ≦ 4. Then write the exponential series for which these approximations are partial sums (perhaps with the
In Problems 1 through 8, apply the successive approximation formula to compute yn (x) for n ≦ 4. Then write the exponential series for which these approximations are partial sums (perhaps with the
In Problems 1 through 8, apply the successive approximation formula to compute yn (x) for n ≦ 4. Then write the exponential series for which these approximations are partial sums (perhaps with the
In Problems 1 through 8, apply the successive approximation formula to compute yn (x) for n ≦ 4. Then write the exponential series for which these approximations are partial sums (perhaps with the
In Problems 1 through 8, apply the successive approximation formula to compute yn (x) for n ≦ 4. Then write the exponential series for which these approximations are partial sums (perhaps with the
In Problems 1 through 8, apply the successive approximation formula to compute yn (x) for n ≦ 4. Then write the exponential series for which these approximations are partial sums (perhaps with the
In Problems 1 through 8, apply the successive approximation formula to compute yn (x) for n ≦ 4. Then write the exponential series for which these approximations are partial sums (perhaps with the
In Problems 1 through 8, apply the successive approximation formula to compute yn (x) for n ≦ 4. Then write the exponential series for which these approximations are partial sums (perhaps with the
In Problems 9 through 12, compute the successive approximations yn (x) for n ≦ 3; then compare them with the appropriate partial sums of the Taylor series of the exact solution. dy dx = x + y, y(0)
In Problems 9 through 12, compute the successive approximations yn (x) for n ≦ 3; then compare them with the appropriate partial sums of the Taylor series of the exact solution. dy dx = y + e*,
In Problems 9 through 12, compute the successive approximations yn (x) for n ≦ 3; then compare them with the appropriate partial sums of the Taylor series of the exact solution. dy dx = y2, y(0) = 1
In Problems 9 through 12, compute the successive approximations yn (x) for n ≦ 3; then compare them with the appropriate partial sums of the Taylor series of the exact solution. dy dx = y³, y(0) =
Apply the iterative formula in (16) to compute the first three successive approximations to the solution of the initial value problem dx dt dy dt = 2x - y, = 3x - 2y, x(0) = 1; y(0) = -1.
Apply the matrix exponential in (19) to solve (in closed form) the initial value problemShow first thatfor each positive integer n.) [i]=0x X, x(0) [B].
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 Euler’s method twice to
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 Euler’s method twice to
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 Euler’s method twice to
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 Euler’s method twice to
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 Euler’s method twice to
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 Euler’s method twice to
A hand-held calculator will suffice for Problems 1 through 10, where an initial value problem and its exact solution are given. Apply the improved Euler method to approximate this solution on the
Separate variables and use partial fractions to solve the initial value problems in Problems 1–8. Use either the exact solution or a computer-generated slope field to sketch the graphs of several
A hand-held calculator will suffice for Problems 1 through 10, where an initial value problem and its exact solution are given. Apply the improved Euler method to approximate this solution on the
A hand-held calculator will suffice for Problems 1 through 10, where an initial value problem and its exact solution are given. Apply the Runge–Kutta method to approximate this solution on the
A hand-held calculator will suffice for Problems 1 through 10, where an initial value problem and its exact solution are given. Apply the improved Euler method to approximate this solution on the
A hand-held calculator will suffice for Problems 1 through 10, where an initial value problem and its exact solution are given. Apply the improved Euler method to approximate this solution on the
A computer with a printer is required for Problems 17 through 24. In these initial value problems, use Euler’s method with step sizes h = 0.1, 0.02, 0.004, and 0.0008 to approximate to four decimal
A computer with a printer is required for Problems 17 through 24. In these initial value problems, use the Runge–Kutta method with step sizes h = 0.2, 0.1, 0.05, and 0.025 to approximate to six
A hand-held calculator will suffice for Problems 1 through 10, where an initial value problem and its exact solution are given. Apply the Runge–Kutta method to approximate this solution on the
A hand-held calculator will suffice for Problems 1 through 10, where an initial value problem and its exact solution are given. Apply the Runge–Kutta method to approximate this solution on the
A hand-held calculator will suffice for Problems 1 through 10, where an initial value problem and its exact solution are given. Apply the improved Euler method to approximate this solution on the
A more detailed version of Theorem 1 says that, if the function f (x, y) is continuous near the point (a, b), then at least one solution of the differential equation y' = f (x, y) exists on some open
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 +
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) = ex
Problems 49 and 50 deal with the solution curves of y'' + 3y' + 2y = 0 shown in Figs. 5.1.6 and 5.1.7.Figure 5.1.7 suggests that the solution curves shown all meet at a common point in the third
A second-order Euler equation is one of the formwhere a, b, c are constants.(a) Show that if x > 0, then the substitution v = ln x transforms Eq. (22) into the constant- coefficient linear
Make the substitution v = ln x of Problem 51 to find general solutions (for x > 0) of the Euler equations in Problems 52–56.x2y'' + xy' - y = 0
Make the substitution v = ln x of Problem 51 to find general solutions (for x > 0) of the Euler equations in Problems 52–56.x2y'' + 2xy' - 12y = 0
Make the substitution v = ln x of Problem 51 to find general solutions (for x > 0) of the Euler equations in Problems 52–56.4x2y'' + 8xy' - 3y = 0
Make the substitution v = ln x of Problem 51 to find general solutions (for x > 0) of the Euler equations in Problems 52–56.x2y'' - 3xy' + 4y = 0
In Problems 15–26, find a basis for the solution space of the given homogeneous linear system. x12x2 + 3x3 = 2x13x2x3 = 0
In Problems 13–18, determine whether the given functions are linearly independent.1 + x, 1 - x, and 1 - x2
In Problems 17–22, three vectors v1, v2, and v3 are given. If they are linearly independent, show this; otherwise find a nontrivial linear combination of them that is equal to the zero vector.v1 =
In Problems 15–26, find a basis for the solution space of the given homogeneous linear system. x1 + 3x2 + 4x3 + 5x4 = 0 2x1 + 6x2 + 9x3 + 5x4 = 0
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, -3, 3, 5), v2 = (2,-5, 9, 3)
In Problems 19–22, use the method of Example 5 to find the constants A, B, and C in the indicated partial-fraction decompositions. x-5 (x-2)(x-3) A x-2 + B x 3
In Problems 15–18, apply Theorem 4 (that is, calculate a determinant) to determine whether the given vectors u, v, and w are linearly dependent or independent.u = (1, 1, 0), v = (4, 3, 1), w = (3,
In Problems 13–18, determine whether the given functions are linearly independent.2 cos x + 3 sin x and 4 cos x + 5 sin x
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. x13x25x3 - 6x4 = 0 2x1 + x2 +
In Problems 17–22, three vectors v1, v2, and v3 are given. If they are linearly independent, show this; otherwise find a nontrivial linear combination of them that is equal to the zero vector.v1 =
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, 5, 2, 3), v2 = (3, 7, 11, 9,5)
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. X5 = 0 x1 + 3x2 + 2x3 +
Let S = {v1, v2, ...., vk} be a basis for the subspace W of Rn. Then a basis T for Rn that contains S can be found by applying the method of Example 5 to the vectorsv1, v2, ......, vk, e1, e2,
Let S = {v1, v2, ...., vk} be a basis for the subspace W of Rn. Then a basis T for Rn that contains S can be found by applying the method of Example 5 to the vectorsv1, v2, ......, vk, e1, e2,
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 3x3 - - 7x4 =
In Problems 9–16, express the indicated vector w as a linear combination of the given vectors v1, v2, ....., vk if this is possible. If not, show that it is impossible.w = (7,7,9, 11); v1 = (2,0,
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, 7, -6, -9)
In Problems 13–16, a set S of vectors in R4 is given. Find a subset of S that forms a basis for the subspace of R4 spanned by S.v1 = (5,4,2,2), v2 = (3,1,2,3), v3 = (7,7,2,1), v4 = (1,-1,2,4), v5 =
In Problems 13–18, determine whether the given functions are linearly independent.cos 2x, sin2 x, and cos2 x
In Problems 15–18, apply Theorem 4 (that is, calculate a determinant) to determine whether the given vectors u, v, and w are linearly dependent or independent.u = (1, -1, 2), v = (3, 0, 1), w = (1,
In Problems 15–26, find a basis for the solution space of the given homogeneous linear system. x13x2 2x15x2 + 2x3 - 4x4 = 0 + 7x3 3x4 = 0
In Problems 15–26, find a basis for the solution space of the given homogeneous linear system. x1 + 3x2 + 4x3 = 0 3x1 + 8x2 + 7x3 =
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, 3, 2, 4), v2 = (2,7,7,3)
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. x1 + 3x2 + 8x3 x4 =
In Problems 17–22, three vectors v1, v2, and v3 are given. If they are linearly independent, show this; otherwise find a nontrivial linear combination of them that is equal to the zero vector.v1 =
In Problems 15–18, apply Theorem 4 (that is, calculate a determinant) to determine whether the given vectors u, v, and w are linearly dependent or independent.u = (5, -2, 4), v = (2, -3, 5), w =
In Problems 13–18, determine whether the given functions are linearly independent.1 + x, x + x2, and 1 - x2
In Problems 23 and 24, use the method of Example 7 to find a basis for the solution space of the given differential equation. (It’s 3-dimensional in Problem 23 and 4-dimensional in Problem 24.)y'''
Given a homogeneous system Ax = 0 of (scalar) linear equations, we say that a subset of these equations is irredundant provided that the corresponding column vectors of the transpose AT are
In Problems 23–26, the vectors {vi} are known to be linearly independent. Apply the definition of linear independence to show that the vectors {ui} are also linearly independent.u1 = v1 + v2, u2 =
In Problems 19–24, use the method of Example 3 to determine whether the given vectors u, v, and w are linearly independent or dependent. If they are linearly dependent, find scalars a, b, and c not
Prove: For arbitrary vectors u and v,(a) |u + v|2 + |u - v|2 = 2|u|2 + 2|v|2;(b) |u + v|2 - |u - v|2 = 4u . v.
In Problems 15–26, find a basis for the solution space of the given homogeneous linear system. x12x2 3x3 - 16x4 – 2x14x2 x12x2 16x4 = 0 + x3 + 17x4 = 0 + 3x3 +26x4: = 0
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, 1, 1, 1, 3), v2 = (2, 3, 1, 4, 7), v3 =
In Problems 17–22, three vectors v1, v2, and v3 are given. If they are linearly independent, show this; otherwise find a nontrivial linear combination of them that is equal to the zero vector.v1 =
Given a homogeneous system Ax = 0 of (scalar) linear equations, we say that a subset of these equations is irredundant provided that the corresponding column vectors of the transpose AT are

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