 3.5.1: In each of 1 through 14, find the general solution of the given dif...
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 3.5.29: Consider the equationy 3y 4y = 2et (i)from Example 5. Recall that y...
 3.5.30: Determine the general solution ofy + 2y = Nm=1am sin mt,where > 0 a...
 3.5.31: . In many physical problems the nonhomogeneous term may be specifie...
 3.5.32: Follow the instructions in to solve the differential equationy + 2y...
 3.5.33: Behavior of Solutions as t . In 33 and 34, we continue the discussi...
 3.5.34: Behavior of Solutions as t . In 33 and 34, we continue the discussi...
 3.5.35: In this problem we indicate an alternative procedure7 for solving t...
 3.5.36: In each of 36 through 39, use the method of to solve the given diff...
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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
Get Full SolutionsElementary 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 stepbystep solutions from this chapter. Chapter 3.5: Nonhomogeneous Equations; Method of Undetermined Coefficients includes 39 full stepbystep 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.

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 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

Inverse matrix AI.
Square matrix with AI A = I and AAl = 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 B1 AI and (AI)T. Cofactor formula (Al)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.

lAII = 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. AI is also symmetric.