Using the method of separation of variables and Fourier series,
solve the following heat
conduction problem in a rod.
∂u/∂t =∂2u/∂x2
, u(0, t) = 0, u(π, t) = 3π, u(x, 0) = 0
Compute, by Euler’s method, an approximate solution to the
following initial value problem for h = 1/8 : y’ = t − y , y(0) = 2
; y(t) = 3e^(−t) + t − 1 . Find the maximum error over [0, 1]
interval.
Use the method of Undetermined Coefficients to find the solution
of the initial value value problem:
y'' + 8y' + 20y = 9cos(2t) - 18e-4t, y(0) = 5. y'(0)
= 0
Use the separation of variables method to solve the following
problem. Consider a long, narrow
tube connecting two large, well-mixed reservoirs containing a
small concentration of N2 in another
inert gas. The tube length is L = 100 cm. To establish an
initial concentration profile in the
tube, each reservoir is held at a fixed concentration:
Reservoir 1 contains no N2 and reservoir 2 has
2 × 10−6 mol/cm3 of N2.
(a) At t = 0, the concentrations of the...
Calculate the Euler method approximation to the solution of the
initial value problem at the given x-values. Compare your results
to the exact solution at these x-values.
y' = y+y^2; y(1) = -1, x = 1.2, 1.4, 1.6, 1.8
Apply Euler's method twice to approximate the solution to the
initial value problem on the interval [0, .5], first with step size
h=0.25, then with step size h= 0.1. Compare the three-decimal place
values of the two approximations at x=.5 with the value of y(.5) of
the actual solution.
y'=y-3x-6, y(0)=8, y(x)=9+3x-ex
a.) The Euler approximation when h=0.25 of y(.5)
is:
b.)The Euler approximation when h=0.1 of y(.5)
is:
Determine the unique solution of the given initial value
problem that is valid in any interval not including the singular
point.
4x2
y’’ + 8xy’ + 17y =
0; y(1)
= 2, y’ (1) = 2(31/2 )− 1
please show all steps
Problem #2:
Use separation of variables with λ = 36 to find a
product solution to the following partial differential
equation,
y
∂2u
∂x2
+
∂u
∂y
= 0
that also satisfies the conditions u(0, 0) = 9 and
ux(0, 0) = 4.