 2.9.1: Determine the Laplace transform of each of the following functions....
 2.9.2: Determine the Laplace transform of each of the following functions.
 2.9.3: Determine the Laplace transform of each of the following functions....
 2.9.4: Determine the Laplace transform of each of the following functions....
 2.9.5: Determine the Laplace transform of each of the following functions.
 2.9.6: Determine the Laplace transform of each of the following functions....
 2.9.7: Determine the Laplace transform of each of the following functions....
 2.9.8: Determine the Laplace transform of each of the following functions....
 2.9.9: Given that L = \/;; /2, find e { t  Hint: Make the change of varia...
 2.9.10: Show that each of the following functions are of exponential order.
 2.9.11: Show that each of the following functions are of exponential order.
 2.9.12: Show that each of the following functions are of exponential order.
 2.9.13: Show that e' does not possess a Laplace transform. Hint: Show that ...
 2.9.14: Suppose that f (t) is of exponential order. Show that F(s)= f2 (f (...
 2.9.15: Solve each of the following initialvalue problems.
 2.9.16: Solve each of the following initialvalue problems.
 2.9.17: Find the Laplace transform of the solution of each of the following...
 2.9.18: Find the Laplace transform of the solution of each of the following...
 2.9.19: Find the Laplace transform of the solution of each of the following...
 2.9.20: Find the Laplace transform of the solution of each of the following...
 2.9.21: Prove that all solutions y (t) of ay" + by' + cy = f (t) are of exp...
 2.9.22: Let F(s) = C { f (t)). Prove that Hint: Try induction.
 2.9.23: Solve the initialvalue problem
 2.9.24: Solve the initialvalue problem y"3y'+2y=e'; y(to)=l, yt(to)=O by...
Solutions for Chapter 2.9: The method of Laplace transforms
Full solutions for Differential Equations and Their Applications: An Introduction to Applied Mathematics  3rd Edition
ISBN: 9780387908069
Solutions for Chapter 2.9: The method of Laplace transforms
Get Full SolutionsDifferential Equations and Their Applications: An Introduction to Applied Mathematics was written by and is associated to the ISBN: 9780387908069. Chapter 2.9: The method of Laplace transforms includes 24 full stepbystep solutions. Since 24 problems in chapter 2.9: The method of Laplace transforms have been answered, more than 5913 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 and Their Applications: An Introduction to Applied Mathematics, edition: 3.

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

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA1 yll2 = Y T(AAT)1 Y = 1 displayed by eigshow; axis lengths ad

Fundamental Theorem.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n  r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

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.

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.

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

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.

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.

Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.

Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.