Engineering Fundamentals: An Introduction to Engineering, 4th ed.c

624 Chapter 18 Mathematics in Engineering

Example 18.18 Evaluate

We use rule 3, a/xdxaln |x| C, from Table 18.7 to solve this problem, as shown.

Example 18.19 Evaluate

We use the substitution method (rule 7) from Table 18.7 to solve this problem, as shown.

For this problem,ux

2

 2 xand du/dx 2 x 2 2(x1), and rearrange the terms as

du2(x1)dxor du/2(x1)dx. Making these substitutions, we get

Example 18.20 Evaluate

We use the substitution method (rule 8), e

u(x)

u(x)dxe

u(x)

C, from Table 18.7 to

solve this problem, as shown. From the previous example,ux

2

 2 xand du/2(x1)dx.

Making these substitutions, we get

18.7 Differential Equations

Many engineering problems are modeled using differential equations with a set of correspond-

ing boundary and /or initial conditions. As the name implies, differential equations contain

derivatives of functions or differential terms. Moreover, the differential equations are derived

by applying the fundamental laws and principles of nature (some of which we described earlier)

to a very small volume or a mass. These differential equations represent the balance of mass,

force, energy, and so on. Boundary conditions provide information about what is happening

physically at the boundaries of a problem. Initial conditions tell us about the initial conditions

of a system (at timet0), before a disturbance or a change is introduced. When possible, the

c1x^12 e

1 x 2  2 x 2

ddx

1

2

(^) e
u
du
1
2
1 e
u
2 C 
1
2
1 e
1 x 2  2 x 2
2 C
c1x^12 e
1 x 2  2 x 2
ddx.

31 x 121 x
2
 2 x 24 dx

u
du
2

1
2

u du
1
2
a
u
2
2
Cb
1
2
c
1 x
2
 2 x 2
2
2
Cd
^31 x^121 x
2
 2 x 24 dx.

10
x
dx10 ln 0 x 0 C

10
x
dx.
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