rdlc barcode font may be solved by adding any particular integral to the complementary function, or in Software

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CHAPTER THREE
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general solution, of the homogeneous equation obtained by replacing E(x) by zero The complementary function may be found from the rules just given And the particular integral may be found assuming that E(x) is a single term or a sum of terms each of which is of the type k, k cos bx, k sin bx, keax, where a and b are any real numbers (See Ref 1 for more details)
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Solutions of Partial Differential Equations The only means of solution that
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will be discussed is the separation-of-variables method For the following equation, 1 z x2
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z y2
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the solution of z z(x, y) will be assumed to have the form of a product of two functions X(x) and Y(y), which are functions of x and y only, respectively: z Substituting z z(x,y) X(x)Y( y)
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XY into the original differential equation, we obtain 2 X x 3X
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2 1 Y g1 2 Y y
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2 1 X 1 2 X x
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Note that the left-hand side contains the function of x only and that the righthand side contains functions of y only Since the right- and left-hand sides are independent of x and y, respectively, they must be equal to a common constant, called a separation constant ; thus, 1 and g1 d 2Y dy2 g2 dY dy [g3 ]Y 0 d 2X dx2 2 dX dx [ 3 ]X 0
Once the solutions to the above have been obtained, the product solution XY is obtained The method outlined may be extended to additional variables and it is applicable also when f1, f2, f3 are functions of x and g1, g2, and g3 are functions of y
LAPLACE TRANSFORMATION
The Laplace transformation of a function (t) is F(s) where (t) s
L[ (t)]
e st (t) dt
a function of a real variable (usually t time) a complex variable of the form ( j )
ENGINEERING MATHEMATICS
F(s)
an equation expressed in the transform variable s, resulting from operating on a function of time with the Laplace integral an operational symbol indicating that the quantity which it pre xes is to be transformed into the frequency domain
Example (t) A (t)
Table 38 lists the transforms of common functions An inverse transformation is represented symbolically as
L 1F(s)
(t)
For any f(t) there is only one direct transform, F(s) For any given F(s) there is only one inverse transform f(t) Therefore, tables are generally used for determining inverse transforms
COMPLEX VARIABLES
1 Complex Numbers A complex number z consists of a real part x and an imaginary part y and is represented as
z where i 1 (i2
iy 1)
The conjugate z of a complex number is de ned as z x iy
Two complex numbers are equal only if their real parts are equal and their imaginary parts are equal; ie, x1 only if x1 Also x only if x 0 and y 0 iy 0 x2 and y1 y2 iy1 x2 iy2
Complex numbers satisfy the distributive, associative, and commutative laws of algebra Complex numbers may be graphically represented on the z(x y) plane or in polar (r, ) coordinates The polar coordinates of a complex number are
CHAPTER THREE
TABLE 38 Laplace Transforms
(t) A f1(t) e 1 e
F(s) A/s F1(s) F2(s) 1 s 1 rs 1 A s s2 s s2
[ (t)]
f2(t)
sin t cos t 1 e e Ae
sin t e
s2 (s
Be C ,B where A a t t t e e e A B C )( where A ( B ( )( C ( )( t t2 tn d / dt[ (t)] d 2 / dt2[ (t)] d 3 / dt3[ (t)] (t)dt 1 sinh t
(s a ,C
1 2 s 1 )(s s a )(s 1
(s ) ) ) 1 s2 2 s3 n! sn 1 sF(s) s2F(s) s2F(s)
(0 )* s (0 ) s2 (0 ) (t)dt 0 df (0 ) dt df (0 ) s dt
d 2 (0 ) dt2
cosh t
1 [F(s) s 1 2 2 s 2 s / (s
* (0 ) initial value of (t), evaluated as t approaches zero from positive values Source: From Avallone and Baumeister1
ENGINEERING MATHEMATICS
r where mod modulus, and
mod z, r
arg z
amp z
where arg argument and amp amplitude arg z is multiple-valued, but for an angular interval of range 2 there is only one value of for a given z
2 Elementary Complex Functions
Polynomials A polynomial in z, anzn an 1zn a0, where n is a positive integer, is simply a sum of complex numbers times integral powers of z which have already been de ned Every polynomial of degree n has precisely n complex roots provided each multiple root of multiplicity m is counted m times Exponential Functions The exponential function ez is de ned by the equation z ex iy ex eiy ex (cos y i sin y) Properties: e0 1; ez1 ez2 e z1 z2 z1 z2 z1 z2 z 2k i z e ; e /e e ;e e Trigonometric Functions sin z (eiz e iz) / 2i; cos z (eiz e iz) / 2; tan z sin z / cos z; cot z cos z / sin z; sec z 1 / cos z; cos z 1 / sin z Fundamental identities for these functions are the same as their real counterparts Thus cos2 z sin2 z 1, cos(z1 z2) cos z1 cos z2 sin z1 sin z2, sin(z1 z2) sin z1 cos cos z1 sin z2 The sine and cosine of z are periodic functions of period 2 ; z2 thus sin(z 2 ) sin z For computation purposes, sin z sin (x iy) sin x i cos x sinh y, where sin x, cosh y, etc, are the real trigonometric and cosh y hyperbolic fucntions Similarly, cos z cos x cosh y i sin x sinh y If x 0 in the results given, cos iy cosh y, sin iy i sinh y
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