2. Show that if
are analytical functions in an environment of the
point
y
so
the equation solutions:
they are also analytical functions in a certain environment of the
same point, what form do the solutions have?
2. Show that if are analytical functions in an environment of the point y so the equation solutions: they are al...
a) By direct substitution determine which of the following functions satisfy the wave equation. 1. g(x, t) = Acos(kx − t) where A, k, are positive constants. 2. h(x, t) = Ae where A, k, are positive constants. 3. p(x, t) = Asinh(kx − t) where A, k, are positive constants. 4. q(x, t) = Ae where A, a, are positive constants. 5. An arbitrary function: f(x, t) = f(kx−t) where k and are positive constants. (Hint: Be careful with...
Consider a second-order linear homogeneous equation
Suppose that
are two solutions. Show that
is also a solution to the equation (plug it in and use the fact
that
and
are solutions).
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consider the variation of constants formula where P(t)= a) show that solves the initial value problem x'+p(t)=(t) x()= when p and q are continuous functions of t on an interval I and tg p(s)ds We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this image tg p(s)ds
First use (20) in Section 6.4.
y'' +
1 − 2a
x
y' +
b2c2x2c − 2 +
a2 − p2c2
x2
y = 0, p ≥
0 (20)
Express the general solution of the given differential equation
in terms of Bessel functions. Then use (26) and (27)
J1/2(x)
=
2
πx
sin(x)
(26)
J−1/2(x)
=
2
πx
cos(x)
(27)
to express the general solution in terms of elementary
functions. (The definitions of various Bessel functions are given
here.)
y''...
find the solution of the inhomogeneous system for y" +p(t)y' +q(t)y = f(t), a second order scalar equation with p, q, f continuous on interval I, for which (to ) = 0, to on I We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this image
Find parametric equation for the line of intersection of the planes Find the point of intersection of a line and line Find an equation of the plane that contains the line and orthogonal to the plane We were unable to transcribe this imagey=0 We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this image
Show that a function , which minimises,
among all smooth functions , s.t. on
, solves
the following equation:
in
and
on
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Find an equation of the tangent plane to the surface f (x, y) =
x tan y at the point (2,
/4, 2).
a. x - 4y - z =
b. None of these
c. x + 4y - z = -
d. -x + 4y - z =
e. - x + 4y - z =
/4
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Let yp(y) be the C(2) inverse demand function facing a monopoly, where y++ is its rate of output, and let yC(y) be the C(2) total cost function of the monopoly. Assume that p(y)>0, p'(y)<0, and C'(y)>0 for all y++, and that a profit maximizing rate of output exists. Total revenue is therefore given by R(y)=p(y)y. Given that question uses an inverse demand function, the elasticity of demand, namely (y), is defined as (y)= 1/p'y p(y)/y. Why is (y)<0? Prove that...
Show that is nowhere dense in if and only if is not an isolated point of . Note: you may need these definitions. The set is said to be nowhere dense in if . Also, is an isolated point of if there exists small enough so that for all . We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this...