MAT1250 Formula Sheet
ALGEBRA
Laws of Arithmetic
ab  ba
Commutative Laws:
Associative Laws:
a  b  c  a  b  c
ab  c  ab  ac
Distributive Law:
ab  ba
abc  abc
Absolute Value
 a , a  0
| a | 
 a, a  0
Fractions
a c ad  cb
, b, d  0
 
b d
bd
a c ac
 
, b, d  0
b d bd
a c a d ad
   
, b, c, d  0
b d b c bc
Inequalities
If a  b then a  c  b  c for any c .
If a  b then ac  bc if c  0 , but ac  bc if c  0 .
If a  b then  a  b .
If a  b then 1  1 .
a
b
If a  b and c  d then a  c  b  d .
Index Laws
For a, b  0 and m, n real:
am
a m a n  a mn
 a m n
an
a 
n m
 a nm
n
n
a a
1

a m  m
 
n
a
b b
( m an integer and n a positive integer)
a0  1
abm  a mbm
m
a n  n am
Logarithm Laws
For a, b, y, m, n  0 and k real
y  a x  loga ( y)  x
loga (1)  0
loga ( a )  1
log a ( mn )  log a ( m )  log a ( n )
m
loga    loga (m)  loga (n)
n
loga (mk )  k loga (m)
b ( m)
loga (m)  log
logb ( a )
Quadratic Equations
 b  b2  4ac
If ax  bx  c  0 then x 
. The term b2  4ac is called the discriminant.
2a
2
Completing the Square:
1
TRIGONOMETRY
General Triangles
A  B  C  180o
sin A sinB  sinC 


a
b
c
2
2
2
a  b  c  2bccos A
1
Area  ab sinC 
2
Right Triangles
a 2  b2  c 2
a
sin A 
c
b
cos A 
c
a sin(A)
tan A  
b cos(A)
Reference Triangles
Angle of 180o is equivalent to an angle of

radians.
Trigonometric Identities
Basic Definitions:
tan  x  
sin x 
cos x 
csc  x  
1
sin  x 
sec  x  
cot  x  
1
cos x 
1
tan x 
Pythagorean Identities :
sin2 x  cos2 ( x)  1
tan2 x   1  sec2 ( x)
1  cot2 x   csc2 ( x)
Odd/Even Properties:
sin x    sinx 
cos x   cosx 
Half-Angle Formulae:
sin2  x  
1  cos2 x 
2
cos2 x  
1  cos2 x 
2
Double-Angle Formulae:
sin2 x   2 sinx cosx 
cos2x   cos2 x   sin2 x 
2
Addition Formulae:
cosx  y   cos xcos y   sinx sin y 
sin x  y   sin x cos y   cos x sin y 
Product Formulae:
1
sinx  y   sinx  y 
2
1
sinx sin y   cos x  y   cosx  y 
2
sin x cos y  
Auxiliary Angle Formula:
cos x cos y  
1
cosx  y   cosx  y 
2
a sinx   b cos x   R sinx    ,
0 

2
b
tan  
a
R 2  a 2  b2 ,
COORDINATE GEOMETRY
Plane
Let P  ( x1, y1 ) and Q  ( x2 , y2 )
Distance between P and Q :
d  ( x2  x1 ) 2  ( y2  y1 ) 2
Gradient of the line through P and Q :
m
Let m1 and m2 be the slopes of two lines:
Lines parallel if m1  m2
Lines perpendicular if m1m2  1 .
Equation of line through P with slope m :
Equation of a circle centred at P with radius r :
Cartesian  polar coordinates:
y2  y1
x2  x1
 y  y1   mx  x1 
x  x1 2   y  y1 2  r 2
x  r cos 
y  r sin 
r  x2  y2
tan  
y
x
Space
Let P  ( x1 , y1 , z1 ) and Q  ( x2 , y2 , z2 )
Distance between P and Q :
d  ( x2  x1 )2  ( y2  y1 )2  ( z2  z1 )2
Equation of a plane:
ax  by  cz  d  0
Equation of a sphere centred at P with radius r :
x  x1 2   y  y1 2  z  z1 2  r 2
MEASUREMENT
Rectangle
Area  xy
Perimeter 2x  2 y
Triangle
Area 
1
bh
2
3
Circle
Area  r 2
Perimeter  2r
Cylinder
Volume  r 2h
SurfaceArea 2r2  2rh
Cone
1
Volume  r 2h
3
Surface Area  r 2  r r 2  h 2
Sphere
4
Volume  r 3
3
SurfaceArea 4r2
FUNCTIONS
Definitions


A function f : A  B is a rule that assigns to each element x in a set A exactly one element f x in a set B .
The domain dom f   A and the range ran f    f  x : x  A  B .
A polynomial is a function of the form f  x  
a x  a  a x  a x    a x
n
i
i
2
0
1
2
n
n
, where ai is real for
i 0
i  0,, n , and n is a non-negative integer.
Px 
, where P and Q are polynomials.
Q x 

A rational function has the form f  x  

A function f is called even if f x  f  x for all x dom f  , and is called odd if f  x   f  x for all

A function f has a global maximum at x  c if f c  f  x for all x dom f  . Similarly, a function f has

x dom f  .
a global minimum at x  c if f c  f  x for all x dom f  .
A function f has a local maximum at x  c if f c  f  x for all x  A , for an open set A  dom f  .
Similarly, a function f has a local minimum at x  c if f c  f  x for all x  A , for an open set
A  dom f  .



A function f : A  B is called one-to-one if for every y B there is no more than one element x  A such
that f  x  y .
A function f : A  B is called onto if for every y B there is at least one element x  A such that f  x  y .
A relation between the variables x and y is called explicit if it has the form y  f  x , otherwise it is called
implicit.
4

A curve defined by x  f t  , y  gt  , t  R , is called a parametric curve with parameter t . The equations

Write lim f ( x )  L to say that as x approaches a from either side, f  x approaches L . Write f  L as

x  a to mean the same thing.
In order for the limit as x approaches a to be defined (i.e. to exist) lim f ( x )  lim f ( x )  lim f ( x ) .
x  f t  , y  gt  are called parametric equations.
xa
x a 
 A function f is continuous at the point x  a if
x a 
x a
lim f ( x )  lim f ( x )  f ( a ) .
x a 
x a
Combinations of Functions
Addition:
Subtraction:
Multiplication:
Division:
 f  g( x)  f ( x)  g( x) ,
 f  g( x)  f ( x)  g( x) ,
 fg( x)  f ( x)g( x) ,
dom ( f  g )  dom ( f )  dom ( g )
dom ( f  g )  dom ( f )  dom ( g )
dom ( fg )  dom ( f )  dom ( g )
f
dom   dom( f )  dom( g )  x : g ( x)  0
g
f
f ( x)
 ( x) 
,
g ( x)
g
f  g( x)  f gx
Composition:
dom f  g   x domg  : gx dom f 
Inverse Functions
1
Let f be a one-to-one function. Then the inverse function f is defined by
f 1 y  x  f x  y
for all y ran f  .
 
 
Domain and Range:
dom f   ran f 1 and dom f 1  ran f 
Cancelation:
f  f 1x  x
f 1  f x  x
 
for all x dom f
1
for all x dom f 
Basic Functions
Power Functions
Odd Powers:
domain =
range =
Even Powers:
domain =
range = {y 
: y  0}
5
Root Functions
Odd Powers:
domain =
range =
Even Powers:
domain = {x  : x  0}
range = {y  : y  0}
Even Powers:
domain = {x  : x  0}
range = {y  : y  0}
Logarithm:
domain = {x 
range =
Reciprocal Power Functions
Odd Powers:
domain = {x  : x  0}
range = {y  : y  0}
Exponential and Logarithm Functions
Exponential:
domain =
range = {y 
: y  0}
: x  0}
6
Cancelation Relationships:
Natural Base:
a loga ( x )  x
ln  x   log e  x 
loga a x   x
for all x  0
for all real x
Trigonometric Functions
sin:
domain =
range = {y 
tan:
domain = {x 
cos:
domain =
range = {y 
( 2n  1)
,n
2
csc:
domain = {x 
( 2n  1)
,n
2
: y  -1 or y  1}
cot:
:  1  y  1}
: x
range = {y 
range =
sec:
domain = {x 
range = {y 
: x
domain = {x 
:  1  y  1}
: x  n , n 
: y  -1 or y  1}
: x  n , n 
range =
7
Inverse Trigonometric Functions
sin-1:
domain = {x 
range = {y 
:  1  x  1}
:

2
 y

2
cos-1:
domain = {x 
range = {y 
}
tan-1:
:  1  x  1}
: 0  y  }
domain =
range = {y 
:

2
 y

2
}
Hyperbolic Functions
Basic Definitions:
sinh x  
Identity:
sinh:
domain =
range =
e x  e x
2
cosh x  
e x  e x
2
tanh  x  
sinh  x 
cosh  x 
cosh2 x   sinh2  x   1
cosh:
domain =
range = {y 
: y  1}
8
tanh:
domain =
range = {y 
:  1  y  1}
Transformations
The transformation of y  f  x  given by
y  af b x  c   d
1
, a vertical translation by
b
represents a vertical scaling by a factor of a , a horizontal scaling by a factor of
d units upward, and a horizontal translation by c units to the right.
Limit Laws
If lim f ( x ) and lim g ( x ) exist, c  R , and n  Z  then
x a
x a
lim c  c
lim x  a
xa
xa
lim  f  x   g  x   lim f  x   lim g  x 
xa
xa
xa
lim f  x g  x   lim f  x lim g  x 
xa

xa

lim f  x   lim f  x 
n
xa
x a
n
xa
lim cf x   c lim f  x 
xa
lim
x a
xa
lim f  x 
f  x  x a

, lim g  x   0
g  x  lim g  x  xa
x a
lim f ( x)  n lim f ( x)
n
x a
x a
If f is continuous at a , and a is not on the boundary of dom f  , then lim f x   f a  .
xa
9
CALCULUS
Derivative Definition
f  x  
df
f x  h   f x 
 lim
h
0
dx
h
Differentiation Rules
Linearity:
If y( x )  af  x   bg x  , a and b real, then
Product Rule:
If y( x)  f ( x)g( x) then
dy
df
dg
a
b
dx
dx
dx
dy df
dg

g x  
f x 
dx dx
dx
df
dg
g x  
f x 
f ( x)
dy dx
dx

then
If y( x) 
2
g ( x)
dx
g x 
Quotient Rule:
dy dy du

dx du dx
Chain Rule:
If y  f u  and u  g  x  then
Parametric Differentiation:
dy
dy dt
dx

 0 then
If x  f t  , y  gt  , and
dx dx
dt
dt
Derivative Related Function Properties

A function f is differentiable at the point x  a if it is continuous at x  a , and lim

The tangent to a curve y  f  x  at x  c has slope f c ; the normal has slope 







xa
df
df
.
 lim
x

a
dx
dx
1
.
f c 
A function f is increasing/decreasing on an interval I  dom f  provided f  x   0 / f x   0 for all x  I .
A function f is concave up/down on the interval I  dom f  provided f  x   0 / f  x   0 for all x  I .
A critical point of the function f is a point x  dom f  such that f  x   0 or f  x  is undefined.
A stationary point of the function f is a point x  dom f  such that f  x   0 .
An inflection point of the function f is a point x  dom f  such that f  x   0 .
If f c   0 and f c   0 , then f has a local minimum at x  c .
If f c   0 and f c   0 , then f has a local maximum at x  c .
Anti-Derivative Definition
A function F(x) is called an anti-derivative of f (x) if
The indefinite integral F ( x ) 

dF
 f (x) .
dx
f ( x ) dx denotes all anti-derivatives of f (x) .
Integration Rules (Indefinite Integrals)
Linearity:
f ax  b  form:
If y  af  x   bg x  , a and b real, then
If F ( x ) 

f  x  dx then


f ax  b  dx 
y dx  a

f ( x ) dx  b

g ( x ) dx .
F ( ax  b )
.
a
10
If f is a differentiable function and g is a continuous function then
Substitution:

f '  x g  f  x  dx 

g ( u ) du where u  f ( x ) .
Definition of Definite Integral
If f is defined for a  x  b , and the interval a, b is divided into n subintervals of equal width
x  b  a  / n . Let x0 , x1 ,, xn be the endpoints of these subintervals and let x1* , x2* ,, xn* be any sample of
*
points in these subintervals such that xi  xi 1 , xi  . Then the definite integral of f from a to b is defined by

b
f  x  dx  lim
n 
a

n
 
f xi* x
i 1
provided that the limit exists.
Integration Rules/Properties (Definite Integrals)
b

Linearity:
c1 f x   c2 gx dx  c1
a
b
b
  
f x dx  c2 g x dx ( c1, c2 , a, b real).
a
b
a
a
( a, b real).
 f x dx    f x dx
 f x dx   f x dx   f x dx
Interchanging Bounds:
a
b
Additivity:
b
c
a
a
b
( a, b, c real).
c
If f is differentiable for x  a, b and g is continuous for x  a, b , and a, b are real, then

b
f '  x g  f  x  dx 
a

f (b)
g (u) du , where u  f ( x ) .
f (a)
Fundamental Theorem of Calculus
If f is continuous on the interval a, b then
1.
d
dx
x
 
f t dt  f (x)
( c real).
c
b
2.

f ( x) dx  F b  F a   F ( x)a where F' ( x)  f ( x)
b
a
Inverse Trigonometric Substitution
Expression
Substitution
Domain
Identity Used
a2  x2
x  a sin  
1  sin 2    cos2  
a2  x2
x  a tan  
  
   , 
 2 2
x2  a2
x  a sec  
  
   , 
1  tan 2    sec 2  
    3 
  0,    , 
sec 2    1  tan 2  
 2 2

2

2 
11
Integration by Parts


df
dg
g  x dx   f  x g  x  
f  x  dx
 dx
 dx
df
g  x dx  f  x g  x  
dx
Indefinite Integral:
f x 
b
dg
dx
dx
b
b
a
Definite Integral:
a
a
Use the following table as a guide:
g x 
df
dx
Polynomial
Exponential
Trigonometric
Hyperbolic
Logarithmic
Inverse Trigonometric
Inverse Hyperbolic
Polynomial
Integration by Partial Fraction Decomposition
Given a rational function
Px 
:
Q (x )
1. If the degree of P  x  is greater than or equal to the degree of Q  x  (i.e. the rational function is
improper), divide P  x  by Q  x  using polynomial long division.
2. If the degree of P  x  is less than the degree of Q  x  then factorise Q  x  completely and use following
table:
Factor in Denominator
ax  b
Terms in Partial Fraction Decomposition
A
ax  b
A1
A2
An


2
ax  b ax  b 
ax  b n
Ax  B
2
ax  bx  c
A1 x  B1
A2 x  B2
An x  Bn



2
2
ax  bx  c ax 2  bx  c 
ax 2  bx  c n
ax  bn
ax
2
 bx  c 
(irreducible, no real roots)
ax

n
 bx  c
(irreducible, no real roots)
2
A useful integral:

1
1
 x
dx  tan 1  
2
a x
a
a
2
Area and Volume formulae
b
 f x  dx
The area between the curves y  f  x  and y  g  x  on the interval x a, b is given by
 f x   g x  dx .
The volume of the solid of revolution of y  f  x  about the x  axis for x a, b is given by V   y dx .

The volume of the solid of revolution of x  g  y  about the y  axis for y  a, b is given by V   x dy .

The area between the curve y  f x  and the x  axis on the interval x a, b is given by
a
b
a
b
2
a
b
2
a
12
Differentiation and integration formulae
dy
Derivative:
dx
Function: y
Integral:

y dx
0
a (constant)
ax  C
nxn1
xn ( n   1 )
x n 1
C
n 1
1
or x1
x
ln x  C
ax lna
ax
ax
C
ln a
ex
ex
ex  C
1
x ln a
1
x
log a x
1
x ln x  x   C
ln a
ln x
x ln x  x  C
cosx
sin x
 cos x  C
 sin x
cosx
sin x  C
tanx
 ln cos x  C
 cot xcsc x
csc x
ln csc x  cot x  C
tan xsec x
sec x
ln sec x  tan x  C
 csc2 x
cot x
ln sin x  C
1
sin1 x
x sin 1 x  1  x 2  C
cos1 x
x cos1 x  1  x 2  C
1
1  x2
tan1 x
1
x tan1 x  ln 1  x 2  C
2
cosh x
sinh x
cosh x  C
sinh x
cosh x
sinh x  C
sec h 2 x
tanh x
lncosh x   C
1
sinh1 x
x sinh 1 x  1  x 2  C
cosh1 x
x cosh1 x  x 2  1  C
tanh1 x
1
x tanh1 x  ln 1  x 2  C
2

1
 x2
2 or
x
sec2 x 
1
cos2 x
1 x

2
1
1 x2
1 x
2
1
x2 1
1
1  x2


13
COMPLEX NUMBERS

Cartesian Form:

Equality:

Addition/Subtraction:

Multiplication:

Conjugate:

Division:

Polar Form:

Cartesian  polar form:
Re(z)  a
a  bi  c  di  a  c and b  d
z  a  bi where i  1 ,
2
Im(z)  b
a  bi  c  di  a  c  b  d i
a  bic  di  ac  bd   ad  bci
If z  a  bi then z  a  bi ,
z1 z1 z2

z2 z2 z2
zz  a 2  b2
z  r cis   rcos   i sin  ,
a  r cos 
r  a2  b2
mod(z )  z  r
arg(z)  
b  r sin 
b
tan   ,   0,2 
a
and z2  r2 cis2  then z1 z2  r1r2 cis1   2 
z
r
and z2  r2 cis2  then 1  1 cis1   2  , r2  0 .
z2 r2

Multiplication (polar):
If z1  r1 cis1 

Division (polar):
If z1  r1 cis1 

De Moivre’s Theorem:
n
n
If z  r cis  then z  r cisn  for n  Z .

Euler’s Relationship:
r cis   rei
VECTORS
For vectors a  a1 , a2   a1i  a2 j and b  b1, b2   b1i  b2 j :
Magnitude:
a  a12  a 22
Unit Vector:
aˆ 
Dot Product:
a  b  a1b1  a2b2
1
a
a
FUNCTIONS OF TWO VARIABLES


A function of two variables is a rule f : A  B is a rule that assigns each element ( x, y) in a set A  R 2
exactly one element f  x, y in a set B  R .
The level curves of a function of two variables z  f x, y  , are the curves with equations f x, y  k , where k
is a constant in the range of f .


If f is a function of two variables x and y , its partial derivatives of f are defined by
f
f
f x  h, y   f x, y 
f x, y  h   f x, y 
 lim
,
 lim
fx 
fy 
h

0
h

0
x
y
h
h
The second partial derivatives of f are
f xx 

2 f

 f x ,

2
x
x
f yy 
2 f

 f y ,

2
y
y
f xy 
2 f

 f x ,

yx y
f yx 
2 f

fy 

xy x
If f is defined on a disk D that contains the point a, b , and f xy and f yx are both continuous on D, then
f xy  f yx .

If f is a function of two variables x and y , then the gradient of f is the vector function f defined by

 f f 
f   ,  . The gradient vector points in the direction of steepest ascent.
 x y 
If f is a function of two variables x and y , then the directional derivative of f in the direction of a unit
vector u is given by Du f  f  u .
14
FIRST ORDER DIFFERENTIAL EQUATIONS
Definitions





A differential equation is an equation that contains an unknown function and one or more of its derivatives.
The order of a differential equation is the order of the highest derivative that occurs in the differential equation.
dy
 f  y .
A first order differential equation is called autonomous if it can be written in the form
dx
dy
 f  x, y 
A first order initial value problem involves finding the particular solution of a differential equation
dx
which also satisfies an initial condition y  x0   y0 .
The interval of validity for the particular solution of an initial value problem is the maximum open interval on
which the particular solution satisfies the differential equation.
Equilibrium Solutions




dy
 f x, y  has an equilibrium solution y x  c if f x, c  0 for all x .
dx
An equilibrium solution is stable if f  x, y  changes sign from positive to negative as y increases past c . A
sufficient condition for y x  c to be stable is f  x, y   0 for y  c , and f  x, y   0 for y  0 .
An equilibrium solution is unstable if f x, y  changes sign from negative to positive as y increases past c . A
sufficient condition for y x  c to be unstable is f  x, y   0 for y  c , and f  x, y   0 for y  0 .
An equilibrium solution is semi-stable if f  x, y  does not changes sign as y increases past c .
A first order differential equation
Separable Differential Equations

A first order differential equation is called separable if it can be written in the form
If f  y   0 then we write

dy

f y

dy
 g x  f  y  .
dx
g  x dx
Linear Differential Equations

A first order differential equation is called linear if it can be written in the form

The general solution of a first order linear differential equation is
y x  
where I  x   e 
P  x dx
1 

I  x  

dy
 P x  y  Q  x  .
dx

I  x Q  x dx  c 

is the integrating factor.
Existence and Uniqueness Theorem
If the functions P  x  and Q x  are continuous on an open interval a  x  b containing the point x  x0 then there
dy
 P x  y  Q x  for all x  a, b , and that also
exists a unique function y x  that satisfies the differential equation
dx
satisfies the initial condition y  x0   y0 where y0 is an arbitrary prescribed initial value.
15
SECOND ORDER DIFFERENTIAL EQUATIONS
Existence and Uniqueness Theorem (EUT)
Consider the second-order linear initial-value problem
d2y
dy
dy
 p x   qx  y  g x  , y x 0   y 0 ,
2
dx
dx
dx
 y0 .
x  x0
If p , q and g are continuous functions on an open interval I that contains the point x  x0 , then there exists one and
only one solution to the differential equation that satisfies the initial conditions.
Fundamental solutions
If y1 and y 2 are solutions of the 2nd-order linear homogeneous DE
d2y
dy
 px   qx  y  0 ,
2
dx
dx
on an interval I , and the Wronskian
W  y1 , y 2  x   y1
dy 2
dy
 y2 1  0
dx
dx
on I , then y1 and y 2 are linearly independent and form a fundamental set of solutions to the DE. The general solution
of the DE is given by
y  x   C1 y1  x   C 2 y 2  x  ,
where C1 and C 2 are constants.
Second-order linear homogeneous differential equations with constant coefficients
d2y
dy
 p  qy  0 , where p and q are constants.
2
dx
dx
2
  p  q  0
General form:
Characteristic equation:
Roots of characteristic equation: 1 and 2
General solution ( C1 and C 2 are constants)
1 and 2 are real but not equal
y  C1e 1 x  C2 e 2 x
1  2  
y  C1e x  C2 xe x
1  2  a  ib
y  C1e ax cosbx   C2 e ax sin bx 
Cauchy-Euler equations
d2y
dy
 bx  cy  0 , where a , b and c are constants.
2
dx
dx
Make the change of variable w  ln x  x  e w , where w is the new independent variable which replaces x .
General form:
ax 2
Then the DE becomes
d2y
dy
 b  a 
 cy  0 ,
2
dw
dw
which is a 2nd-order linear DE with constant coefficients.
a
16
Second-order linear nonhomogeneous DE’s
Given the nonhomogeneous DE
d2y
dy
 p x   qx  y  g x  ,
2
dx
dx
Solution strategy:
Find the general solution y c  x   C1 y1  C 2 y 2 of the corresponding homogeneous DE
1.
2.
d2y
dy
 px   qx  y  0
2
dx
dx
This solution is called the complementary solution.
Find some single solution y p  x  of the nonhomogeneous DE. This solution is called a particular solution.
3.
Add together these two functions to obtain the general solution of the nonhomogeneous DE.
Method of undetermined coefficients for DE’s with constant coefficients
Right hand side: g x 
Choice for y p x 
an  x   an x n  an 1 x n 1  ...  a1 x  a0
An  x   An x n  An 1 x n 1  ...  A1 x  A0
a n  x ex
An  x ex
an x sin  x  or a n  x cos x 
An  x sin  x   Bn x cos x 
An x sin x   Bn x cosx ex
an  x e sin  x  or an  x e cosx 
x
x
Determine the unknown constants by equating coefficients.
Method of variation of parameters
Given the nonhomogeneous DE
1.
2.
3.
4.
5.
d2y
dy
 p x   qx  y  g x  ,
2
dx
dx
Find the general solution y c  x   C1 y1 x   C 2 y 2  x  of the corresponding homogeneous DE.
du
y  x g  x 
du 2
y  x g  x 
Let 1   2
and
.
 1
dx
W  y1 , y 2  x 
dx W  y1 , y 2  x 
Integrate these two equations to find u1  x  and u2 x  , omitting the integration constants.
A particular solution is y p  x   u1  x  y1 x   u2  x  y2  x  .
The general solution is y  x   y c  x   y p  x  .
LAPLACE TRANSFORMS

Defintion:
Behaviour of F s  as s   :
L f t    F s  

0
h
lim F s   lim L  f t    0
s

e  st f t dt  lim e  st f t dt
h 
0
s
Solution of an Initial Value Problem (IVP) via the method of Laplace Transforms
d2y
dy
dy
 p t   qt  y  g t  , y 0   y 0 ,
 y 0' can be solved as follows:
2
dt
dt
dx t  0
1. Denote the Laplace Transform of y(t ) by Y ( s) .
2. Take the Laplace Transform of both sides of the DE to convert it into algebraic equation in s and Y ( s) .
3. Solve the equation from Step 2. for Y ( s) .
4. Evaluate L-1 Y s    y t  to obtain the solution y(t ) to the IVP.
The IVP
17
Table of Laplace Transforms
Laplace Transform F s   L f t  
Function f t   L-1  F s  
1
, s0
s
n!
, s  0 , n  1,2,3, 
s n 1
1
, sa
sa
s
, s0
2
s  a2
a
,s 0
2
s  a2
s
, s a
2
s  a2
a
, s a
2
s  a2
π
, s0
2s 3 / 2

, s0
s
1
tn
e at
cosat 
sinat 
cosh at 
sinh at 
1
t2
t
 12
Shifting theorem 1: e at f t 
F s  a 
Shifting theorem 2: uc t  f t  c 
e  cs F s 
f t 
sF s   f 0 
f t 
s 2 F s   sf 0   f 0 
t n f t 
 1n F ( n ) ( s )
0 , t  c
, c0
uc t   
1 , t  c
e  cs
s
f t  , f t  T   f t 
1
1  e Ts
T

e  st f t dt
0

0
 t  t0   
f * g (t ) 

,
t  t0
, otherwise
t
0
f (t  u )g (u )du 

e  st0
t
f (u )g (t  u )du
F ( s )G ( s )
0
18