If F(s) is the Laplace transform of f(t) , denoted by F(s) =L{ f(t)} , find the
inverse Laplace transform L−1{F(as + b)} in terms of f(t) , where a > 0 and
b ≠ 0 .
Solve
Let
Compute eAt .
Consider the non-homogeneous linear system x´= Ax + eαtv , where x is a vector
consisting of functions in t , α is not an eigenvalue of A , and v ≠ 0 is a constant
vector. Find a particular solution of the system, in terms of A,α , v and t .
(1) Determine the Fourier series coefficients (an , bn ) of the function f(t) = t⋅u(t)
expanded over the interval (−π,2π) , where u(t) is the unit-step function.
(2) If the coefficients (an ,bn ) from (1) are also the Fourier series coefficients of
some function expanded over the interval (−2π, 4π) , find the function in terms of
f(t) .
Solve the following boundary value problem for f(x,t) with x > 0 and 0 < t < 10
Solve the differential equation (x2 −1) y´´− 6xy´ +12y = 0 by power series of the form
(1) Find the recurrence relation of cn .
(2) Find the two linearly independent solutions. Write the first three nonzero terms of
each series if it is an infinite series.
(3) Find the guaranteed radius of convergence.
A system of linear equations has unknown coefficients which can be expressed with
the real variable a . The system is as follows
Please determine the value of a for the system to have nontrivial solutions.
Given n× n positive definite matrices A and B , for any x ≠ 0 , derive the
Find the expression for x , or function thereof, to achieve this minimum value.
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