 7.7.1.292: In 1 and 2 use the definition of the Laplace transform to find .f(t...
 7.7.1.293: In 1 and 2 use the definition of the Laplace transform to find .f(t...
 7.7.1.294: In 324 fill in the blanks or answer true or false.If f is not piece...
 7.7.1.295: In 324 fill in the blanks or answer true or false.The function f(t)...
 7.7.1.296: In 324 fill in the blanks or answer true or false. F(s) s2(s2 4) is...
 7.7.1.297: In 324 fill in the blanks or answer true or false. If and , then
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 7.7.1.302: In 324 fill in the blanks or answer true or false.{t sin 2t}
 7.7.1.303: In 324 fill in the blanks or answer true or false.{sin 2t (t )}
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 7.7.1.305: In 324 fill in the blanks or answer true or false. 1 1 3s 1
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 7.7.1.309: In 324 fill in the blanks or answer true or false. 1 e 5s s2
 7.7.1.310: In 324 fill in the blanks or answer true or false. 1 s s2 2 e s
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 7.7.1.313: In 324 fill in the blanks or answer true or false.{te8t { f(t)} F(s...
 7.7.1.314: In 324 fill in the blanks or answer true or false.{ f(t)} F(s)and k...
 7.7.1.315: In 324 fill in the blanks or answer true or false.{eat f(t k)(t k)}...
 7.7.1.316: In 2528 use the unit step function to find an equation for each gra...
 7.7.1.317: In 2528 use the unit step function to find an equation for each gra...
 7.7.1.318: In 2528 use the unit step function to find an equation for each gra...
 7.7.1.319: In 2528 use the unit step function to find an equation for each gra...
 7.7.1.320: In 2932 express f in terms of unit step functions. Find and .1 1 23...
 7.7.1.321: In 2932 express f in terms of unit step functions. Find and .f(t) 3...
 7.7.1.322: In 2932 express f in terms of unit step functions. Find and .1 2 3 ...
 7.7.1.323: In 2932 express f in terms of unit step functions. Find and .1 2 1 t f
 7.7.1.324: In 3340 use the Laplace transform to solve the given equation.y 2y ...
 7.7.1.325: In 3340 use the Laplace transform to solve the given equation.y 8y ...
 7.7.1.326: In 3340 use the Laplace transform to solve the given equation.y 6y ...
 7.7.1.327: In 3340 use the Laplace transform to solve the given equation.y 5y ...
 7.7.1.328: In 3340 use the Laplace transform to solve the given equation.where...
 7.7.1.329: In 3340 use the Laplace transform to solve the given equation.y 5y ...
 7.7.1.330: In 3340 use the Laplace transform to solve the given equation.y (t)...
 7.7.1.331: In 3340 use the Laplace transform to solve the given equation. t 0 ...
 7.7.1.332: In 41 and 42 use the Laplace transform to solve each system.x y t 4...
 7.7.1.333: In 41 and 42 use the Laplace transform to solve each system.x y e2t...
 7.7.1.334: The current i(t) in an RCseries circuit can be determined from the...
 7.7.1.335: A series circuit contains an inductor, a resistor, and a capacitor ...
 7.7.1.336: A uniform cantilever beam of length L is embedded at its left end (...
 7.7.1.337: When a uniform beam is supported by an elastic foundation, the diff...
 7.7.1.338: (a) Suppose two identical pendulums are coupled by means of a sprin...
 7.7.1.339: Coulomb Friction Revisited In in Chapter 5 in Review we examined a ...
 7.7.1.340: Range of a ProjectileNo Air Resistance (a) A projectile, such as th...
 7.7.1.341: Range of a ProjectileWith Air Resistance (a) Now suppose that air r...
Solutions for Chapter 7: The Laplace Transform
Full solutions for Differential Equations with BoundaryValue Problems,  8th Edition
ISBN: 9781111827069
Solutions for Chapter 7: The Laplace Transform
Get Full SolutionsDifferential Equations with BoundaryValue Problems, was written by and is associated to the ISBN: 9781111827069. This textbook survival guide was created for the textbook: Differential Equations with BoundaryValue Problems,, edition: 8. This expansive textbook survival guide covers the following chapters and their solutions. Since 50 problems in chapter 7: The Laplace Transform have been answered, more than 22956 students have viewed full stepbystep solutions from this chapter. Chapter 7: The Laplace Transform includes 50 full stepbystep 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!

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

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

Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

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].

Cyclic shift
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

Dimension of vector space
dim(V) = number of vectors in any basis for V.

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

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

Graph G.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n  1)/2 edges between nodes. A tree has only n  1 edges and no closed loops.

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.

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.

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.

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn 1 with P(Xi) = bi. Vij = (Xi)jI and det V = product of (Xk  Xi) for k > i.