MAT1137 and MAT1236: Formula Sheet ALGEBRA
Laws of Arithmetic
a b ba a b c a b c ab c ab ac
Commutative Laws: Associative Laws: Distributive Law:
ab ba abc abc
Absolute Value
a , a 0 a, a 0
| a | Fractions a
c
d c
ad cb
, b, d 0 bd ac , b, d 0 b d bd a c a d ad , b, c, d 0 b d b c bc
b a
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:
a a a m n
mn
am a
n
n
an a n b b
a
m
a
n m
a m n
abm a m b m m
1 a
a nm
a 1 0
m
an
n
am
( m an integer and n a positive integer) Logarithm Laws For a, b, y, m, n
0 and k real
y a x loga ( y) x
loga (1) 0
loga (a) 1
1
m loga (m) loga (n) n
loga (m)
loga (mk ) k loga (m)
loga
loga (mn) loga (m) loga (n) logb ( m) logb ( a)
Roots of Quadratic Equations 2
If ax
bx c 0 then
x
b b 2 4ac 2a
. The term b
4ac is called the discriminant.
2
TRIGONOMETRY
General Triangles
A B C 180o sin A a
sin B
b
sin C c
a2 b2 c2 2bc cos A Area
1 2
ab sin C
Right Triangles
a2 b2 c2 sin A cos A
tan A
a c b c a
b
sin( A) cos( A)
Reference Triangles
o Angle of 180 is equivalent to an angle of radians.
Trigonometric Identities Basic Definitions:
tan x
sin x cos x
csc x
1 sin x
sec x
1 cos x
cot x
1 tan x
2
m loga (m) loga (n) n
loga (m)
loga (mk ) k loga (m)
loga
loga (mn) loga (m) loga (n) logb ( m) logb ( a)
Roots of Quadratic Equations 2
If ax
bx c 0 then
x
b b 2 4ac 2a
. The term b
4ac is called the discriminant.
2
TRIGONOMETRY
General Triangles
A B C 180o sin A a
sin B
b
sin C c
a2 b2 c2 2bc cos A Area
1 2
ab sin C
Right Triangles
a2 b2 c2 sin A cos A
tan A
a c b c a
b
sin( A) cos( A)
Reference Triangles
o Angle of 180 is equivalent to an angle of radians.
Trigonometric Identities Basic Definitions:
tan x
sin x cos x
csc x
1 sin x
sec x
1 cos x
cot x
1 tan x
2
Pythagorean Identities :
sin 2 x cos2 ( x) 1
tan 2 x 1 sec2 ( x)
1 cot 2 x csc2 ( x )
Odd/Even Properties: sin x sin x
cos x cos x
Half-Angle Formulae:
sin2 x
1 cos2 x
cos2 x
2
1 cos2 x 2
Double-Angle Formulae:
cos2 x cos2 x sin 2 x
sin 2 x 2 sin x cos x
Addition Formulae: sin x y sin x cos y cos x sin y
cos x y cos x cos y sin x sin y
Product Formulae:
sin x cos y sin x sin y
1 2 1 2
sin x y sin x y
cos x cos y
1 2
cos x y cos x y
cos x y cos x y a sin x b cos x R sin x ,
Auxiliary Angle Formula:
0
tan
R 2 a 2 b 2 ,
2
b a
COORDINATE GEOMETRY
Plane Let P ( x1, y1) and Q ( x2 , y2 )
Distance between P and Q :
d
Gradient of the line through P and Q :
m
Let m1 and m2 be the slopes of two lines:
Lines parallel if m1
Lines perpendicular if m1m2
Equation of a circle centred at P with radius r : Cartesian
y2 y1 x2 x1
m2
1 .
Equation of line through P with slope m :
( x2 x1 ) 2 ( y2 y1 ) 2
polar coordinates:
y y1 m x x1 x x12 y y12 r 2 x r cos y r sin r x 2 y 2
tan
y x 3
Space
Let P ( x1 , y1 , z 1 ) and Q ( x2 , y2 , z 2 ) Distance between P and Q :
d
( x2 x1 ) 2 ( y2 y1 ) 2 ( z 2 z 1 ) 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 z 1 2 r 2
MEASUREMENT
Rectangle
Area xy Perimeter 2 x 2 y
Triangle Area
1 2
bh
Circle
Area r 2 2 r Perimeter
Cylinder
Volume r 2 h Surface Area 2 r 2 2 rh
Cone Volume
1 3
r 2 h
Surface Area r 2 r r 2 h 2
Sphere Volume
4 3
r 3
Surface Area 4 r 2
4
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 r an f f x : x A B . n
A polynomial is a function of the form f x ai x i a0 a1 x a2 x 2 an x n , where ai is real i 0
for i
0,, n , and n is a non-negative integer.
A rational function has the form f x
P x Q x
, where P and Q are polynomials.
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 x dom f .
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 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. A curve defined by x f t , y g t , t R , is called a parametric curve with parameter t . The equations x f t , y g t are called parametric equations. Write lim f ( x ) L to say that as x approaches a from either side, f x approaches L . Write x a
f L as
a to mean the same thing.
In order for the limit as approaches a to be defined (i.e. to exist) lim f ( x ) lim f ( x ) lim f ( x ) . x a
x a
x a
A function f is continuous at the point x a if lim f ( x ) lim f ( x ) f ( a ) . x a
x a
5
Combinations of Functions Addition:
f g ( x) f ( x) g ( x) ,
dom( f g ) dom( f ) dom( g )
Subtraction:
f g ( x) f ( x) g ( x) ,
dom( f g ) dom( f ) dom( g )
Multiplication: fg ( x) f ( x) g ( x) ,
Division:
Composition: f g ( x ) f g x
dom( fg ) dom( f ) dom( g )
f f ( x ) ( x) , ( ) g g x
f dom( f ) dom( g ) x : g ( x) 0 g
dom
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
for all x dom f
f 1 f x x
for all x dom f
1
Basic Functions Power Functions
Odd Powers:
domain R range R
Even Powers: domain R
range y R : y 0
6
Root Functions
Odd Powers:
domain R range R
Even Powers: domain x R:x 0
range y R : y 0
Reciprocal Power Functions
Odd Powers:
domain x R : x 0 range y R : y 0
Even Powers: domain x R : x 0
range y R : y 0
7
Exponential and Logarithm Functions
Exponential:
domain R range y R:y 0
Logarithm:
range R
Cancelation Relationships:
x for all x 0 log a a x x for all real x
Natural Base:
ln x loge x
a
domain x R:x 0
log a ( x )
Trigonometric Functions
sin:
domain R range y R: 1 y 1
cos:
domain R range y R: 1 y 1
8
tan:
domain x R:x
(2n 1) 2
, z Z
cosec: domain x R:x n , z Z
range y R: y 1 or y 1
range R
sec:
(2n 1) , z Z 2 range y R: y 1 or y 1
domain x R:x
cot:
domain x R:x n , z Z range R
9
Inverse Trigonometric Functions
-1
sin :
tan-1:
x 2 2 range y R: 1 y 1
domain x R :
domain x R : range R
-1
cos :
domain x R : 0 x range y R: 1 y 1
x 2 2
10
Hyperbolic Functions Basic Definitions:
sinh x
e x e x 2
Identity:
sinh:
cosh x
tanh x
2
sinh x cosh x
cosh 2 x sinh 2 x 1
domain R
cosh:
range R
tanh:
e x e x
domain R range y R : y 1
domain R range y R : 1 y 1
Transformations The transformation of y f x given by y af b x c d
represents a vertical scaling by a factor of a , a horizontal scaling by a factor of
1
b translation by d units upward, and a horizontal translation by c units to the right.
, a vertical
11
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
lim cf x c lim f x
x a
x a
lim f x g x lim f x lim g x
x a
x a
x a
x a
f x
x a g x
x a
lim f x lim f x n
lim
x a
n
lim f x
x a
lim g x
,
x a
lim g x 0
x a
lim n f ( x) n lim f ( x)
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
L’Hospital’s Rule:
Suppose that f and g are differentiable and g x 0 on an open interval I that contains a (except possibly at a ). Suppose that lim f x 0 lim g x , or lim f x lim g x
x a
Then lim
f x
x a g x
x a
x a
is an indeterminate form of type
lim
f x
x a g x
lim
x a
0 0
or
, and
f x
x a g x
if the limit on the right hand side exists (or is or
).
12
CALCULUS
Derivative Definition f x
df dx
lim
f x h f x
h0
h
Differentiation Rules dy
Linearity:
If y af x bg x , a and b real, then
Product Rule:
If y f ( x) g ( x) then
Quotient Rule:
df dg g x f x f ( x) dy dx dx If y then 2 g ( x) dx g x
Chain Rule:
If y f u and u g x then
dy dx
df dx
g x
dy dx
dg dx
dx
a
df dx
b
dg dx
f x
dy du du dx
dy Parametric Differentiation:
If x f t , y g t , and
dx dt
0 then
dy dx
dt dx
dt Derivative Related Function Properties A function f is differentiable at the point x a if it is continuous at x a , and lim x a
The tangent to a curve y f x at x c has slope f c ; the normal has slope
df dx
1 f c
lim x a
df dx
.
.
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 .
13
Anti-Derivative Definition A function F ( x) is called an anti-derivative of f ( x) if
dF dx
f ( x) .
The indefinite integral F ( x ) f ( x ) dx denotes all anti-derivatives of f ( x) . Integration Rules (Indefinite Integrals) Linearity:
If y af x bg x , a and b real, then y dx a f ( x ) dx b g ( x ) dx
Substitution:
If f is a differentiable function and g is a continuous function then
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 n
f x dx lim f x x b
n
a
* i
i 1
provided that the limit exists. Integration Rules/Properties (Definite Integrals) b
b
b
a
a
a
c1 f x c2 g xdx c1 f xdx c2 g xdx
Linearity:
Interchanging Bounds:
b
b
a
Additivity:
a
a
f x dx f x dx
( a, b real).
b
f x dx
c
a
f x dx
( c1 , c2 , a , b real).
b
f xdx c
( a, b, c real).
If f is differentiable for x a, b and g is continuous for x a , b , and a, b are real, then
b
a
f ' x g f x dx
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. 2.
d dx
b
a
x
f t dt f ( x) c
( c real). b
f ( x) dx F b F a F ( x)a where F ' ( x) f ( x) 14
Inverse Trigonometric Substitution Expression
Substitution
Domain
Identity Used
a 2 x 2
x a sin
1 sin 2 cos 2
a 2 x 2
x a tan
, 2 2 , 2 2
x 2 a2
x a sec
3 , 2 2
sec 2 1 tan 2
1 tan 2 sec 2
0,
Integration by Parts
df dg g x dx f x g x f x dx dx dx
Indefinite Integral:
Definite Integral:
b df
a
g x dx f x g x a dx b
b
a
f x
dg dx dx
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
P x
Given a rational function
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
Terms in Partial Fraction Decomposition
ax b
A ax b A1
ax bn
ax b
A2
ax b 2
(irreducible, no real roots) 2
An
ax b n
Ax B
ax2 bx c
ax
bx c
n
(irreducible, no real roots)
A1 x B1 ax 2 bx c
ax2 bx c A2 x B2
ax
2
bx c
2
An x Bn
ax
2
bx c
n
15
Another Useful Integral (Partial Fractions):
a
1 2
x
dx 2
x tan 1 a a 1
Area and Volume formulae The area between the curve y f x and the x axis on the interval x a, b is given by
b
f x dx a
The area between the curves y f x and y g x on the interval x a, b is given by b
f x g x dx . a
The volume of the solid of revolution of y f x about the x axis for x a, b is given by V
b
2 f ( x ) dx . a
The volume of the solid of revolution of x g y about the y axis for y a, b is given by
b
V g ( y ) dy . 2
a
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. A first order differential equation is called autonomous if it can be written in the form
dy f y . dx
A first order initial value problem involves finding the particular solution of a differential equation dy f x, y which also satisfies an initial condition y x0 y0 . dx 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 A first order differential equation all
dy dx
f x, y has an equilibrium solution y x c if f x, c 0 for
.
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 . 16
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 . 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 dx
g x f y .
dy
f y g xdx Linear Differential Equations A first order differential equation is called linear if it can be written in the form
dy dx
P x y Q x .
The general solution of a first order linear differential equation is y x
where I x e
P x dx
1
I x
I xQ x dx c
is the integrating factor.
Exact Differential Equations The differential equation M ( x, y )dx N ( x, y )dy 0 is called exact if
M N . The general y x
solution of an exact differential equation is
f ( x, y ) 0 where
f M ( x, y )dx N ( x, y )dy 0 Existence and Uniqueness Theorem 1:
Consider the initial-value problem dy dx If f and
f x, y , y x0 y0 .
f are continuous functions on the rectangular region a x b , c y d y
containing the point x0 , y0 , then in some interval x0 h x x0 h contained in a x b there exists one and only one solution to the differential equation that satisfies the initial condition. 17
Theorem 2:
Consider the initial-value problem dy dx
P x y Q x , y x0 y0 .
If P and Q 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 condition. SOME SPECIAL FIRST-ORDER DEs
Homogeneous equations General form M x, y dx N x, y dy 0 , where M tx, ty t n M x, y and N tx, ty t n N x, y . Use the change of variable y ux if N x, y is less complicated than M x, y , where u u x is the new dependent variable which replaces y y x and dy xdu udx . Use the change of variable x vy if M x, y is less complicated than N x, y , where v v y is the new independent variable which replaces x and dx ydv vdy . The resulting differential equation is separable. Bernoulli equations General form dy P x y Q x y n . dx 1 n For n 0,1 we make the change of variable w y where w w x is the new dependent variable which replaces y y x . Then dw 1 n P x w 1 nQ x , dx which is a linear differential equation. SECOND-ORDER DIFFERENTIAL EQUATIONS
Existence and Uniqueness Theorem:
Consider the second-order linear initial-value problem d 2 y dy p x q x y g x , y x0 y0 , dx 2 dx
dy dx x x0
y0 .
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. 18
Fundamental solutions nd
If y1 and y 2 are solutions of the 2 -order linear homogeneous DE d 2 y dx
2
p x
dy dx
q x y 0 ,
on an interval I , and the Wronskian y1 y , y W 1 2 x dy1
y2 dy dy dy2 y1 2 y2 1 dx dx dx dx is not zero on an interval I , then y1 and y2 are linearly independent and form a fundamental set of solutions to the DE. The general solution of the DE is given by y x C 1 y1 x C 2 y2 x ,
where C 1 and C 2 are constants. Reduction of order Suppose y1 is a solution of d 2 y
p x
dx 2 Let y x v x y1 x to obtain the first-order DE y1
where w
dv dx
dy dx
q x y 0 .
dy 2 1 py1 w 0 , dx dx
dw
. Then y vy1 y1 w x dx is the general solution y x C 1 y1 x C 2 y2 x .
Second-order linear homogeneous differential equations with constant coefficients d 2 y
p
dy
qy 0 , where p and q are constants. dx 2 dx 2 Characteristic equation: p q 0 General form:
Roots of characteristic equation: 1 and 2
General solution
1 and 2 are real but not equal
y C 1e
1 2
y C 1e x C 2 xe x
1 2 a ib
y C 1e ax cosbx C 2eax sinbx
1 x
C 2e x 2
Cauchy-Euler equations 2 dy 2 d y General form: ax 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 . d 2 y dy Then the DE becomes a b a cy 0 ,which is a 2nd-order linear DE with constant coefficients. 2 dw dw 19
Second-order linear nonhomogeneous DE’s 1.
Find the general solution yc x C 1 y1 C 2 y2 of the corresponding homogeneous DE. This solution is
2.
called the complementary solution. Find some single solution y p x of the nonhomogeneous DE. This solution is called a particular
3.
solution. Add together these two functions to obtain the general solution of the nonhomogeneous DE.
Method of undetermined coefficients for DE’s with constant coefficients g x
Choice for y p x
a n x a n x n a n1 x n1 ... a1 x a0
An x An x n An 1 x n1 ... 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
a n x e x sin x or a n x e x cos x
An x sin x Bn x cos xe x
Determine the unknown constants by equating coefficients. Variation of parameters Given the nonhomogeneous DE d 2 y dx 2
p x
dy dx
q x y g x ,
1.
Find the general solution yc x C 1 y1 x C 2 y2 x of the corresponding homogeneous DE.
2.
Let
3.
Integrate these two equations to find u1 x and u2 x , omitting the integration constants.
4.
A particular solution is y p x u1 x y1 x u2 x y2 x .
5.
The general solution is y x yc x y p x .
du1 dx
y2 x g x W y1 , y2 x
and
du 2 dx
y1 x g x W y1 , y2 x
.
RECTILINEAR MOTION
Let denote displacement, v denote velocity, a denote acceleration, s denote speed, and d denote distance travelled between t t 1 and t t 2 . Then: v
dx dt
x vt dt
a
dv dt
d 2 x
s v(t )
dt 2
v a(t ) dt
The average velocity v on the interval t 1 t t 2 is given by v
d
t 2
s(t ) dt t 1
xt 2 xt 1 t 2 t 1 20
SYSTEMS OF 2 FIRST-ORDER DIFFERENTIAL EQUATIONS
A solution vector of a system of linear first order differential equations X (t ) A (t ) X(t ) F (t ) where
p11 (t ) p12 (t ) g 1 (t ) x (t ) ( ) ( ) A G t t , and X (t ) p (t ) p (t ) g (t ) 22 y (t ) 21 2 is any vector that satisfies X (t ) A (t ) X (t ) G(t ) on I. Existence and Uniqueness Theorem:
Let each of the functions p11, p21, p12, p22, g 1, and g 2, be continuous on an open interval I : a t b , let t 0 be any point in I and let x0 and y0 be any given numbers. Then there exists a unique solution of the system dx p11 (t ) x p12 (t ) y g 1 (t ) dt dy p21 (t ) x p22 (t ) y g 2 (t ) dt that also satisfies the initial conditions x(t 0 ) x0 and y (t 0 ) y0 . Moreover, the solution exists on the interval I .
Homogeneous systems A system of 2 first order differential equations X (t ) A (t ) X(t ) F (t ) is homogeneous if F(t ) 0 ’
x1i A set x i i , 1 , 2 of 2 linearly independent solution vectors of X (t ) A (t ) X (t ) is called a x 2i
fundamental set of solutions on I .
x1i The set x i , i 1,2 is linearly independent on I if and only if the Wronskian x 2 i W ( x 1 , x 2 ) det[ x 1 (t ) x 2 ( t ) ] 0 on I. The general solution of X (t ) A (t ) X (t ) is given by X (t ) c1x 1 (t ) c2 x 2 (t ) . Non-homogeneous systems 4.
Find the general solution Xc t C 1x1 (t ) C 2 x 2 (t ) of the corresponding homogeneous system of DEs.
5.
Find some single solution x p t of the nonhomogeneous system of DEs. This solution is called a
6.
particular solution. Add together these two functions to obtain the general solution of the nonhomogeneous system of DEs.
Variation of parameters
x11 (t ) x 21 (t ) x1i where Define (t ) , i 1 , 2 x i is a fundamental set of solutions. x t x t ( ) ( ) x 22 12 2i
1 Then x p (t ) (t ) v( t ) is a solution of the non-homgeneous DE where v(t ) (t )g(t ) dt
. 21
Homogeneous systems of 2 first order linear DEs with constant coefficients Here X ( t ) AX (t ) where A is a 2×2 matrix. d X
Ax dt Calculate the eigenvalues 1 and 2 of A by finding the solutions of Solution process for the system
a det c
1 0 0 1 (a )(d ) bc 0 . d b
1) If the eigenvalues are real and distinct, determine the corresponding eigenvectors v 1 and v 2 by
solving ( A i I ) v i
0 , then write the general solution as x(t ) c1e t v1 c2e t v 2 2) For equal eigenvalues determine the eigenvector v 1 by solving ( A 1 I ) v1 0 and the generalised t t eigenvector by ( A 1 I ) w v 1 then write the general solution as x(t ) c1e v1 c2 e (t v1 w) 3) If the eigenvalues are complex conjugates, determine the eigenvector for 1 i by solving ( A 1 I ) v1 0 . The vector v 1 is complex: v1 a bi with a and b real. Then write the general t t solution as x(t ) c1e (a cos vt b sin vt ) c2 e (a sin vt b cos vt ) 1
2
1
1
4) If initial conditions are given, solve for c1 and c2 to obtain the specific solution for the IVP. Stability properties of x Ax , detA=ad-bc 0 Eigenvalues 1 2 0
Type of equilibrium point Node
Stability Unstable
1 2 0
Node
Asymptotically Stable
1 0 2
Saddle Point
Unstable
1 2 0
Proper or improper node
Unstable
1 2 0
Proper or improper node
Asymptotically Stable
1 i 2 0 0
Spiral point
1 i 2
Center
Unstable Asymptotically Stable Stable
Non-homogeneous systems with constant coefficient matrix Method of undetermined coefficients gt
Choice for x p t
a n t a n t n a n1t n1 ... a1t a 0
A n t A n x n A n 1t n 1 ... A1t A 0
a n t e t
A n t e t
a n t cos t or a n t sin t
A n t sin t Bn t cos t
a n t e x sin t or a n t e x cos t
A n t sin t Bn t cos t e t 22
TAYLOR SERIES
The Taylor series of a function f about x a is
f x
f n a n!
n 0
a
x a n f a f a( x a) f
2
a
x a 2 f
6
x a 3
The Maclaurin series of a function f is
f n 0 n f 0 2 f 0 3 f x x f 0 f 0 x x x ! 2 6 n n 0
The nth degree Taylor polynomial T n,a x of the function f about
T n,a x
n
f i a i!
i 0
x a f a f a( x a) i
f a 2
a is given by
2
x a
f n a n!
x an
The remainder function Rn,a x is defined by Rn,a x f x T n,a x . Taylor’s Formula: If f has n 1 derivatives in an interval I that contains the number a , then for x in I there is a number z strictly between x and a such that the remainder function can be expressed as Rn,a x
f n1 z
n 1!
x a n1
The Taylor series T n ,a of a function f converges to f at x (i.e. T n,a x f x ) if lim Rn,a ( x) 0 . n
The radius of convergence R of a Taylor series T n,a of a function f , is the maximum value of R such that T n,a x f x as n for all x a R, a R .
Useful limit: lim
x n
n
n!
0
Important Maclaurin series and their radius of convergence 1 1 x
x
e
1 x x2 x3
R 1
n 0
x
n
x n
x 2
x3
n! 1 x 2 6
R
n0
sin x
1
n
2n 1!
n0
cos x
1 n 0
x 2n1
n
x 2n
2n!
x
1
x 2 2
x3 6
x 4 24
x5 120
x6 720
x7 5040
R R 23
1
tan x
1
n
n0
x 2 n1 2n 1
x
x3
3
x5 5
x7
R 1
7
FOURIER SERIES
Theorem: Suppose that f and f are piecewise continuous on the interval L x L . Further, suppose that f is defined outside the interval L x L so that it is periodic with period 2 L . Then f has a Fourier Series
f x
a0
2
m x m x am cos bm sin L L m1
where
m x dx, L L L 1 L m x bm f x sin dx, L L L am
1
L
f x cos
(i)
m 0,1,2, m 1,2,
If L x L and f has both left and right derivatives at x then the Fourier series F of f at x converges to
F ( x ) 12 [ f ( x ) f ( x )] where f ( x ) lim y x, y x f ( y) and
f ( x ) lim y x , y x f ( y) . (ii)
If the right derivative of f exists at x L and the left derivative of f exists at x L , then at both
x L and x L the Fourier series F of f converges to F ( L) F ( L) 12 [ f ( L ) f ( L)]
Even/Odd Properties: If f is an even function then its Fourier series coefficients are given by
am
2 L
m x dx, L
L
f x cos
0
bm 0,
m 0,1,2,
m 1,2,
If f is an odd function then its Fourier series coefficients are given by
am 0, bm
2
m 0,1,2,
m x dx, L
L
L
f x sin
0
m 1,2,
Fourier Cosine/Sine Series: If f is defined on the half interval 0 x L then its Fourier Cosine series is given by
f x
m x am cos 2 m 1 L
a0
24
am
2 L
m x dx, L
L
0
f x cos
m 0,1,2,
and its Fourier Sine series is given by
f x
m x L
bm sin
m 1
bm
2
L
m x dx, L
L
0
f x sin
m 1,2,
SOLUTIONS TO SELECTED BOUNDARY VALUE PROBLEMS Problem y y 0, 0 x p
y (0) y ( p ) 0 y y 0, 0 x p y(0) y( p) 0 y y 0, 0 x p y (0) y( p) 0 y y 0, 0 x p y(0) y ( p) 0 y y 0, x p y ( p) y ( p) y( p) y( p)
Eigenvalue
n n n n
Eigenfunction
n 2
2
p 2 n 2 2 p 2
, n 1
yn ( x) sin
, n 0
yn ( x ) cos
2n 12 2 4 p 2
2n 12 2
n
4 p 2 n 2 2 p 2
, n 1
yn ( x ) sin
, n 1
yn ( x ) cos
, n 0
n x p n x p
( 2n 1) x 2 p ( 2n 1) x
y1n ( x ) sin y2 n ( x) cos
2 p n x p n x p
PARTIAL DIFFERENTIAL EQUATIONS
Linear second order partial differential equation
Au xx Bu xy Cu yy Du x Eu y Fu G( x, y ) Separation of variables: Substitute u ( x, y ) X ( x )Y ( y ) into the partial differential equation and separate the variables to obtain two ordinary differential equations in the variables x and y.
25
COMPLEX NUMBERS
Cartesian Form:
z a bi where i 2 1 ,
Equality:
a bi c di
Addition/Subtraction:
a bi c di a c b d i
Multiplication:
a bi c di ac bd ad bc i
Conjugate:
z a 2 b 2 If z a bi then z a bi , z
z 1
Division:
z 2
Polar Form:
Cartesian
polar form:
Re( z ) a
Im( z ) b
a c and b d
z 1 z 2 z 2 z 2
z r cis r cos i sin ,
a r cos
b r sin
r a2 b2
tan
mod( z ) z r
arg( z )
b
Multiplication (polar):
, 0,2 a If z 1 r 1 cis 1 and z 2 r 2 cis 2 then z 1 z 2 r 1r 2 cis 1 2
Division (polar):
If z 1 r 1 cis 1 and z 2 r 2 cis 2 then
De Moivre’s Theorem:
n n If z r cis then z r cisn for n Z .
Roots:
If z r cis and n Z then 1
1
2 k , n
z n r n cis
Sets in the Argand Plane:
z 2
r 1 r 2
cis 1 2 , r 2 0 .
k 0,1, 2,, n 1
Line - z : a Im( z ) b Re( z ) 1 Circle - z : z P R
Euler’s Relationship:
z 1
( P the circle centre, R the radius)
r cis rei
26
VECTORS 2
Vectors in R
y
Cartesian x, y to polar r , :
r x 2 y 2
tan
Polar r , to Cartesian x, y :
x r cos
y r sin
x
For vectors a a1, a2 a1i a2 j and b b1, b2 b1i b2 j : Addition and Subtraction:
a b a1 b1 , a 2 b2
Scalar Multiplication:
k a ka1 , ka 2 where k is a scalar.
Magnitude:
a
Unit Vector:
aˆ
Dot Product:
a b a1b1 a2b2
a12 a 22
aa
1 a a
a b a b cos where is the angle between a and b . 3
Vectors in R
For vectors a a1, a2 , a3 a1i a2 j a3k and b b1, b2 , b3 b1i b2 j b3k : Addition and Subtraction:
a b a1 b1 , a 2 b2 , a3 b3
Scalar Multiplication:
k a ka1 , ka 2 , ka3 where k is a scalar.
Magnitude:
a
a12 a 22 a32
1
aa
Unit Vector:
aˆ
Dot Product:
a b a1b1 a 2 b2 a3b3
a
a
a b a b cos where is the angle between a and b .
Cross Product:
a b a 2 b3 a3b2 , a3b1 a1b3 , a1b2 a 2 b1 || a b || a b sin where is the angle between a and b . a b b a
Projections For vectors in either R 2 or R 3: 27
v b b b b projbˆ v v bˆ bˆ projb v
Projection of v onto b :
compb v v bˆ
Component of v along b : Relative Displacement and Velocity
r r A r B .
The displacement/position of A relative to B :
A B
The velocity of A relative to B :
A
v B v A v B .
VECTOR CALCULUS
Vector Valued Function Definition A vector valued function is a function whose domain is a subset of n
R
. A function whose domain and range are both subsets of
R is
R,
and whose range is a subset of
called a real valued function.
Differentiation
d r
Definition:
dt
rt lim h 0
rt h r t
h
Rule:
If rt f t i g t j ht k then rt f t i g t j ht k
Unit Tangent:
Tt
rt rt
Integration Indefinite Integral:
f t i g t j ht k dt f t dt i g t dt j ht dt k
Definite Integral:
f t dt i g t dt j ht dt k f t g t h t dt i j k b
b
a
a
b
a
b
a
Curvilinear Motion Let r denote displacement, v denote velocity, a denote acceleration, and s denote speed. Then: v t r t
rt vt dt
at vt r t
s t v(t ) r(t )
vt a(t ) dt
28
The speed s t of an object will be increasing if s t 0 , and deceasing if s t 0 . Alternatively, the speed of an object will be increasing if v t at 0 , and deceasing if vt at 0 . These results only apply if st 0 . Uniform Circular Motion An object moving in uniform circular r t A cos t b i A sin t b j .
motion
in
2
R has
displacement
given
by
Projectile Motion Assuming that the only force acting on an object moving in R 2 is that of gravity, then at 0, g , where g is the acceleration due to gravity. 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 x h, y f x, y lim x h 0 h f f x, y h f x, y f y lim y h 0 h f x
The second partial derivatives of f are
2 f f x f xx x 2 x 2 f f y f yy y 2 y 2 f f f xy y x y x 2 f f f yx x y x y 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 .
29
If f is a function of two variables x and y , then the gradient of f is the vector function f
f f , . The gradient vector points in the direction of steepest ascent. x y
defined by f
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 . A function f of two variables
and y , has a critical point at x, y a, b if
f x a, b 0 f y a, b or if one of the partial derivatives does not exist. Suppose that all of the second partial derivatives of f are continuous on a disk with center a, b and that f x a, b 0 and f y a, b 0 , and define the Hessian matrix
f xxa, b f yxa, b H a, b , , f a b f a b xy yy Then
If det H a, b 0 and f xx a, b 0 , then f a, b is a local minimum. If det H a, b 0 and f xx a, b 0 , then f a, b is a local maximum. If det H a, b 0 , then f a, b is a saddle point of f . MATRICES
Row Operations Consider a system of linear equations in augmented matrix form A | b , where Ri is the i th row, R j is the j th row, and k is a scalar. The following elementary row operations (EROs) can be applied:
( i th row interchanged with j th row)
Ri R j
th th Ri Ri kR j ( k times the j row added to the i row)
Ri kRi
( i th row multiplied by k )
Matrix Operations For m n matrices A and B , n p matrix C and scalar k where:
a11 a12 a a22 21 A am1 am2
a1n
amn a2n
b11 b12 b b22 21 B bm1 bm 2
b1n
bmn b2 n
c11 c12 c c22 21 C cn1 cn 2
c1 p
c2 p
cnp 30
then
ka11 ka12 ka ka22 21 kA kam1 kam 2
ka1n
kamn ka2n
a11 b11 a12 b12 a b a22 b22 21 21 A B am1 bm1 am2 bm 2
a11c11 a12c21 a1ncn1 a11c12 a12c22 a1ncn 2 a c a c a c a c a c a c 21 11 22 21 2 n n1 21 12 22 22 2n n2 AC am1c11 am2c21 amncn1 am1c12 am2c22 amncn 2 a11 a21 a a22 12 T A a1n a2n
am1
am 2
a1n b1n
amn bmn a2 n b2n
a11c1 p a12c2 p a1ncnp
a21c1 p a22c2 p a2ncnp
am1c1 p am 2c2 p amncnp
amn
Algebraic Properties For m n matrices A , B and D , n p matrices C and E , and p q matrix F : Commutative Law:
A B B A
Associative Law:
A B D A B D
Distributive Law:
AC E AC AE
AC F ACF
Identity Matrix The identity matrix is a square matrix with ones along the main diagonal entries and zeros in all other entries. 1 0 0
Property of the Identity:
0 1 0 I 0 0 1 AI IA A for any matrix A and appropriately sized identity.
Matrix Inverse If A and B are square matrices such that AB BA I , then A is said to be invertible and B is called the 1
inverse of A (denoted A ).
For
a
22
matrix
a11 A a21
a12
a22
the
inverse
A 1
a 22 det( A) a 21 1
a12 where a11
det( A) a11 a 22 a12 a 21 is called the determinant of A . If det( A) 0 then A is not invertible (i.e. A is singular). 31
1
For a general n n matrix A , use EROs to convert the augmented matrix A | I to the form I | A singular then this is not possible.
. If A is
32
Differentiation and integration formulas
dy
y
y dx
a (constant)
ax C
dx 0
nx
1
x
2
n 1
x n 1
x ( n 1 ) n
or x 2 x
1
x
or x 1
a
e x
ex log a x
x ln a 1
C
ln x C
a x
x
a ln a
1
n 1
C
ln a
e x C 1 ln a
x ln x x C
ln x
x ln x x C
cos x
sin x
cos x C
sin x
cos x
sin x C
sec 2 x
tan
x
cot x cosec x
cosec x
lnsec x C ln cosec x cot x C
tan x sec
sec
lnsec x tan x C
cosec 2 x
cot
lnsin x C
1
1 x 1
2
1 x 1
1
2
2
sin 1 x
x sin 1 x 1 x 2 C
cos 1 x
x cos 1 x 1 x 2 C
tan 1 x
x tan x 1
1 2
ln1 x 2 C
cosh x
sinh x
cosh x C
sinh x
cosh x
sinh x C
sinh 1 x
x sinh 1 x 1 x 2 C
cosh 1 x
x cosh 1 x x 2 1 C
1 1 x 1
2
x 1 1 2
1 x
2
MAT2437 Differential Equations Formula Sheet
tanh 1 x
x tanh 1 x
1 2
ln1 x 2 C
14