Q1. The exact solution to the boundary value problem

, and  

for y(4) is

-234.67

0.0000

16.000

37.333

Q2. Given

, , ,

the value of  at y(4) using the finite difference method and a step size of h=4 can be approximated by

Q3. Given

, , ,

 The value of y(4) using the finite difference method with a second order accurate central divided difference method and a step size of h=4 is

 0.000

37.333

-234.67

-256.00

Q4. The transverse deflection u of a cable of length, L, that is fixed at both ends, is given as a solution to

where

T = tension in cable

R = flexural stiffness

q = distributed transverse load

Given L=50", T=200 lbs, q=75lbs/in, R=75x10⁶ lbs-in², using finite difference method modeling with second order central divided difference accuracy and a step size of h=12.5", the value of the deflection at the center of the cable most nearly is

0.072737"

0.08832"

0.081380"

0.084843"

Q5. The radial displacement, u of a pressurized hollow thick cylinder (inner radius=5″, outer radius=8″) is given at different radial locations.

The maximum normal stress, in psi, on the cylinder is given by

The maximum stress, in psi, with second order accuracy is

Hint:     ,

 and

where

2079.3

2104.5

2130.7

2182.0

Q6. For a simply supported beam (at x=0 and x=L) with a uniform load q, the vertical deflection v(x) is described by the boundary value ordinary differential equation as

            ,

where

        E = Young’s modulus of elasticity of beam

        I = second moment of area.

This ordinary differential equation is based on assuming that dv/dx is small.  If dv/dx is not small, then the ordinary differential equation is given by