The first six Legendre polynomials are
Find the first three positive values of λ for which the problem
(1− x2 ) y′′ − 2xy′ +λy = 0
y(0) = 0 , y(x) , y′(x) bounded on [−1,1]
has nontrivial solutions (solutions other than y(x) = 0 ).
The square error of a function F relative to a function f on the interval
Suppose we want to minimize the square error of a function F = a + bsin x ( a and
b are constants) relative to f(x) = x +π (−π < x <π ) , what are the best choices of
constants a and b that give the smallest square error?
Solve 2y′′ + ty′ − 2y = 10 , y(0) = y′(0) = 0
(1) Find the Laplace transform of the differential equation.
(2) Solve the 1st-order equation from (1).
(3) Solve y(t) by finding the inverse Laplace transform of the solution in (2).
Solve
u(x,0) = e−2x , x > 0
Find the eigenvalues and eigenfunctions of the boundary value problem
y′′+λy = 0 , y′(0) = 0 , y′(L) = 0
The set B = {u1, u2 , u3} , where
is the basis of R3 . Transform B into an orthonormal basis B" .
Use the inverse of the matrix A to solve the system AX = B ,
where R is the region bounded by the graphs of y = x , x = 2 , and y = 0 by
means of the change of variables x = u + uv , y = v + uv .
Let f(z) = zng(z) , where n is a positive integer, g(z) is entire, and g(z) ≠ 0
for all z . Let C be a circle with center at the origin.
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