 2.2.2.1.43: In 122 solve the given differential equation by separation of varia...
 2.2.2.1.44: In 122 solve the given differential equation by separation of varia...
 2.2.2.1.45: In 122 solve the given differential equation by separation of varia...
 2.2.2.1.46: In 122 solve the given differential equation by separation of varia...
 2.2.2.1.47: In 122 solve the given differential equation by separation of varia...
 2.2.2.1.48: In 122 solve the given differential equation by separation of varia...
 2.2.2.1.49: In 122 solve the given differential equation by separation of varia...
 2.2.2.1.50: In 122 solve the given differential equation by separation of varia...
 2.2.2.1.51: In 122 solve the given differential equation by separation of varia...
 2.2.2.1.52: In 122 solve the given differential equation by separation of varia...
 2.2.2.1.53: In 122 solve the given differential equation by separation of varia...
 2.2.2.1.54: In 122 solve the given differential equation by separation of varia...
 2.2.2.1.55: In 122 solve the given differential equation by separation of varia...
 2.2.2.1.56: In 122 solve the given differential equation by separation of varia...
 2.2.2.1.57: In 122 solve the given differential equation by separation of varia...
 2.2.2.1.58: In 122 solve the given differential equation by separation of varia...
 2.2.2.1.59: In 122 solve the given differential equation by separation of varia...
 2.2.2.1.60: In 122 solve the given differential equation by separation of varia...
 2.2.2.1.61: In 122 solve the given differential equation by separation of varia...
 2.2.2.1.62: In 122 solve the given differential equation by separation of varia...
 2.2.2.1.63: In 122 solve the given differential equation by separation of varia...
 2.2.2.1.64: In 122 solve the given differential equation by separation of varia...
 2.2.2.1.65: In 2328 find an explicit solution of the given initialvalue proble...
 2.2.2.1.66: In 2328 find an explicit solution of the given initialvalue proble...
 2.2.2.1.67: In 2328 find an explicit solution of the given initialvalue proble...
 2.2.2.1.68: In 2328 find an explicit solution of the given initialvalue proble...
 2.2.2.1.69: In 2328 find an explicit solution of the given initialvalue proble...
 2.2.2.1.70: In 2328 find an explicit solution of the given initialvalue proble...
 2.2.2.1.71: In 29 and 30 proceed as in Example 5 and find an explicit solution ...
 2.2.2.1.72: In 29 and 30 proceed as in Example 5 and find an explicit solution ...
 2.2.2.1.73: In 3134 find an explicit solution of the given initialvalue proble...
 2.2.2.1.74: In 3134 find an explicit solution of the given initialvalue proble...
 2.2.2.1.75: In 3134 find an explicit solution of the given initialvalue proble...
 2.2.2.1.76: In 3134 find an explicit solution of the given initialvalue proble...
 2.2.2.1.77: (a) Find a solution of the initialvalue problem consisting of the ...
 2.2.2.1.78: Find a solution of that passes through the indicated points.
 2.2.2.1.79: Find a singular solution of 21. Of 22.
 2.2.2.1.80: Show that an implicit solution of is given by ln(x2 10) csc y c. Fi...
 2.2.2.1.81: Often a radical change in the form of the solution of a differentia...
 2.2.2.1.82: Often a radical change in the form of the solution of a differentia...
 2.2.2.1.83: Often a radical change in the form of the solution of a differentia...
 2.2.2.1.84: Often a radical change in the form of the solution of a differentia...
 2.2.2.1.85: Every autonomous firstorde equation dydx f(y) is separable. Find e...
 2.2.2.1.86: (a) The autonomous firstorder differential equation dydx 1(y 3) ha...
 2.2.2.1.87: In 4550 use a technique of integration or a substitution to find an...
 2.2.2.1.88: In 4550 use a technique of integration or a substitution to find an...
 2.2.2.1.89: In 4550 use a technique of integration or a substitution to find an...
 2.2.2.1.90: In 4550 use a technique of integration or a substitution to find an...
 2.2.2.1.91: In 4550 use a technique of integration or a substitution to find an...
 2.2.2.1.92: In 4550 use a technique of integration or a substitution to find an...
 2.2.2.1.93: (a) Explain why the interval of definition of the explicit solution...
 2.2.2.1.94: (a) If a 0, discuss the differences, if any, betweenthe solutions o...
 2.2.2.1.95: In 43 and 44 we saw that every autonomous firstorder differential ...
 2.2.2.1.96: (a) Solve the two initialvalue problems: and (b) Show that there a...
 2.2.2.1.97: Find a function whose square plus the square of its derivative is 1.
 2.2.2.1.98: (a) The differential equation in is equivalent to the normal form i...
 2.2.2.1.99: Suspension Bridge In (16) of Section 1.3 we saw that a mathematical...
 2.2.2.1.100: (a) Use a CAS and the concept of level curves to plot representativ...
 2.2.2.1.101: (a) Find an implicit solution of the IVP (b) Use part (a) to find a...
 2.2.2.1.102: (a) Use a CAS and the concept of level curves to plot representativ...
Solutions for Chapter 2.2: FirstOrder Differential Equations
Full solutions for Differential Equations with BoundaryValue Problems,  8th Edition
ISBN: 9781111827069
Solutions for Chapter 2.2: FirstOrder Differential Equations
Get Full SolutionsSince 60 problems in chapter 2.2: FirstOrder Differential Equations have been answered, more than 20488 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Differential Equations with BoundaryValue Problems,, edition: 8. Chapter 2.2: FirstOrder Differential Equations includes 60 full stepbystep solutions. Differential Equations with BoundaryValue Problems, was written by and is associated to the ISBN: 9781111827069.

Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

Jordan form 1 = M 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b  Ax is orthogonal to all columns of A.

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

Lucas numbers
Ln = 2,J, 3, 4, ... satisfy Ln = L n l +Ln 2 = A1 +A~, with AI, A2 = (1 ± /5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.

Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Ql. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

Pascal matrix
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

Pivot.
The diagonal entry (first nonzero) at the time when a row is used in elimination.

Reflection matrix (Householder) Q = I 2uuT.
Unit vector u is reflected to Qu = u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q1 = Q.

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!