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# Solutions for Chapter 3.5: Nonhomogeneous Equations; Method of Undetermined Coefficients

## Full solutions for Elementary Differential Equations | 10th Edition

ISBN: 9780470458327

Solutions for Chapter 3.5: Nonhomogeneous Equations; Method of Undetermined Coefficients

Solutions for Chapter 3.5
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##### ISBN: 9780470458327

Elementary Differential Equations was written by and is associated to the ISBN: 9780470458327. Since 39 problems in chapter 3.5: Nonhomogeneous Equations; Method of Undetermined Coefficients have been answered, more than 12232 students have viewed full step-by-step solutions from this chapter. Chapter 3.5: Nonhomogeneous Equations; Method of Undetermined Coefficients includes 39 full step-by-step solutions. This textbook survival guide was created for the textbook: Elementary Differential Equations, edition: 10. This expansive textbook survival guide covers the following chapters and their solutions.

Key Math Terms and definitions covered in this textbook
• Basis for V.

Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

• Complex conjugate

z = a - ib for any complex number z = a + ib. Then zz = Iz12.

• Cross product u xv in R3:

Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

• Hilbert matrix hilb(n).

Entries HU = 1/(i + j -1) = Jd X i- 1 xj-1dx. Positive definite but extremely small Amin and large condition number: H is ill-conditioned.

• Inverse matrix A-I.

Square matrix with A-I A = I and AA-l = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B-1 A-I and (A-I)T. Cofactor formula (A-l)ij = Cji! detA.

• Iterative method.

A sequence of steps intended to approach the desired solution.

• Kirchhoff's Laws.

Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

• lA-II = l/lAI and IATI = IAI.

The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n - 1, volume of box = I det( A) I.

• Left nullspace N (AT).

Nullspace of AT = "left nullspace" of A because y T A = OT.

• Minimal polynomial of A.

The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A - AI) if no eigenvalues are repeated; always meA) divides peA).

• Nullspace N (A)

= All solutions to Ax = O. Dimension n - r = (# columns) - rank.

• Pivot.

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

• Plane (or hyperplane) in Rn.

Vectors x with aT x = O. Plane is perpendicular to a =1= O.

• Rank r (A)

= number of pivots = dimension of column space = dimension of row space.

• Semidefinite matrix A.

(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

• 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!

• Special solutions to As = O.

One free variable is Si = 1, other free variables = o.

• Spectral Theorem A = QAQT.

Real symmetric A has real A'S and orthonormal q's.

• Stiffness matrix

If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

• Symmetric matrix A.

The transpose is AT = A, and aU = a ji. A-I is also symmetric.

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