CHE 527 Advanced Applied Mathematical Analysis in Chemical Engineering Fall 2006
Homework 5
Due in class on Thursday, October 5.
1. Find the complete solution of
(x2+ 1) d2y
dx2−2xdy
dx + 2y= 6(x2+ 1)2
using variation of parameters. Note that the homogeneous problem was solved as part of the previous
homework.
2. Find the complete solution to
dy
dx =3x2y−y3
3xy2−x3
3. Find the general solution of
yd2y
dx2+dy
dx 2
= 0
4. Consider the horizontal motion of the mass/spring system show below:
mass 1 mass 2
x x
1 2
Let x1and x2be the horizontal displacement of the two masses from static equilibrium. Let x1= 1 f t,
x2= 0, ˙x1= 0, and ˙x2= 0 at t= 0, and neglect friction and the mass of each spring. The ‘dot’ (i.e.,
˙x) denotes a time derivative.
The matlab files ‘hw5num4.m’ and ‘springs.m’ solve this problem using forward Euler. Compare the
errors for 100, 1000, and 1 ×104time steps. Is the error O(h) as expected? Why? Also solve the
problem using 4th-order Runge-Kutta.
5. For a thick slab, initially at temperature To, with the temperature of its face suddenly raised to T1at
time t= 0, use the similarity solution from class to find the heat flux (q=−k∂T
∂y ) as a function of
time.
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