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Copyright © WeSolveThem LLC 1 The Ultimate Cheat Sheet for Math & Physics Preview Table of Contents Algebra .............................................................................................................................................. 2 Arithmetic...................................................................................................................................................................................................... 2 Exponents ...................................................................................................................................................................................................... 3 Trigonometry ..................................................................................................................................... 4 Double Angle Formulas ........................................................................................................................................................................... 4 Half Angle Formulas.................................................................................................................................................................................. 5 Sum and Difference Formulas .............................................................................................................................................................. 6 Precalculus ......................................................................................................................................... 7 Equation of a Line ...................................................................................................................................................................................... 7 Equation of Parabola ................................................................................................................................................................................ 7 Equation of Circle ....................................................................................................................................................................................... 7 Equation of Ellipse ..................................................................................................................................................................................... 7 Equation of Hyperbola ............................................................................................................................................................................. 7 Equation of Hyperbola ............................................................................................................................................................................. 7 Calculus .............................................................................................................................................. 8 Tangent line .................................................................................................................................................................................................. 8 Implicit differentiation ............................................................................................................................................................................ 9 Linear Algebra .................................................................................................................................. 10 Rank of matrix and pivots ................................................................................................................................................................... 10 Length of a vector and the unit vector ........................................................................................................................................... 11 Solutions of Augmented Matrices .................................................................................................................................................... 12 Coefficient Matrix .................................................................................................................................................................................... 12 Unique Solution ....................................................................................................................................................................................... 13 Infinite Solution ....................................................................................................................................................................................... 13 No Solution................................................................................................................................................................................................. 13 Differential Equations ...................................................................................................................... 14 First-Order Linear Non-Homogeneous.......................................................................................................................................... 14 Order and Linearity ................................................................................................................................................................................ 14 Reduction of Order ................................................................................................................................................................................. 15 Physics ............................................................................................................................................. 16 Vectors ......................................................................................................................................................................................................... 16 Dot Product ................................................................................................................................................................................................ 16 Cross Product ............................................................................................................................................................................................ 16 Magnitude or Length of a vector ...................................................................................................................................................... 17 Resultant Vector ...................................................................................................................................................................................... 17 Quick Reference ............................................................................................................................... 19 Arithmetic................................................................................................................................................................................................... 19 Exponential ................................................................................................................................................................................................ 19 Radicals ....................................................................................................................................................................................................... 19 Fractions ..................................................................................................................................................................................................... 19 Logarithmic................................................................................................................................................................................................ 19

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TheUltimateCheatSheetforMath&PhysicsPreview

TableofContentsAlgebra..............................................................................................................................................2Arithmetic......................................................................................................................................................................................................2Exponents......................................................................................................................................................................................................3

Trigonometry.....................................................................................................................................4DoubleAngleFormulas...........................................................................................................................................................................4HalfAngleFormulas..................................................................................................................................................................................5SumandDifferenceFormulas..............................................................................................................................................................6

Precalculus.........................................................................................................................................7EquationofaLine......................................................................................................................................................................................7EquationofParabola................................................................................................................................................................................7EquationofCircle.......................................................................................................................................................................................7EquationofEllipse.....................................................................................................................................................................................7EquationofHyperbola.............................................................................................................................................................................7EquationofHyperbola.............................................................................................................................................................................7

Calculus..............................................................................................................................................8Tangentline..................................................................................................................................................................................................8Implicitdifferentiation............................................................................................................................................................................9

LinearAlgebra..................................................................................................................................10Rankofmatrixandpivots...................................................................................................................................................................10Lengthofavectorandtheunitvector...........................................................................................................................................11SolutionsofAugmentedMatrices....................................................................................................................................................12CoefficientMatrix....................................................................................................................................................................................12UniqueSolution.......................................................................................................................................................................................13InfiniteSolution.......................................................................................................................................................................................13NoSolution.................................................................................................................................................................................................13DifferentialEquations......................................................................................................................14First-OrderLinearNon-Homogeneous..........................................................................................................................................14OrderandLinearity................................................................................................................................................................................14ReductionofOrder.................................................................................................................................................................................15

Physics.............................................................................................................................................16Vectors.........................................................................................................................................................................................................16DotProduct................................................................................................................................................................................................16CrossProduct............................................................................................................................................................................................16MagnitudeorLengthofavector......................................................................................................................................................17ResultantVector......................................................................................................................................................................................17

AlgebraArithmetic

𝑎𝑏 ± 𝑎𝑐 = 𝑎 𝑏 ± 𝑐 = 𝑏 ± 𝑐 𝑎 10± 6 = 2 ∙ 5± 2 ∙ 3 = 2 5± 3 = 5± 3 2_________________________________________________________________________________________________________________

_

𝑎𝑏𝑐 =

𝑎𝑏𝑐

123 =

1231=12 ∙13 =

12 ∙ 3 =

16

__________________________________________________________________________________________________________________

𝑎𝑏 ±

𝑐𝑑 =

𝑎𝑑 ± 𝑏𝑐𝑏𝑑

12±

34 =

1 ∙ 4± 2 ∙ 32 ∙ 4 =

4± 68

__________________________________________________________________________________________________________________

𝑎 − 𝑏𝑐 − 𝑑 =

𝑏 − 𝑎𝑑 − 𝑐

1− 23− 4 =

−(−1+ 2)−(−3+ 4) =

2− 14− 3

__________________________________________________________________________________________________________________

𝑎𝑏 + 𝑎𝑐

𝑎 = 𝑏 + 𝑐,𝑎 ≠ 0 12± 16

4 =124 ±

164 = 3± 4

__________________________________________________________________________________________________________________

𝑎𝑏𝑐 =

𝑎𝑏𝑐

165 =

4 ∙ 45 = 4

45

__________________________________________________________________________________________________________________

𝑎𝑏𝑐=

𝑎1 ∙

𝑐𝑏 =

𝑎𝑐𝑏

234

=2134

=21 ∙43 =

83

__________________________________________________________________________________________________________________

𝑎 ± 𝑏𝑐 =

𝑎𝑐 ±

𝑏𝑐

12± 165 =

125 ±

165

__________________________________________________________________________________________________________________

𝑎𝑏𝑐𝑑

=𝑎𝑏 ∙𝑑𝑐 =

𝑎𝑑𝑏𝑐

1234

=12 ∙43 =

46 =

23

__________________________________________________________________________________________________________________

𝑖𝑓 𝑎 ± 𝑏 = 0 𝑡ℎ𝑒𝑛 𝑎 = ∓𝑏 𝑥 ± 2 = 0 ⇒ 𝑥 = ∓2

Exponents

𝑎! = 𝑎 2 = 2!_________________________________________________________________________________________________________________

_

𝑎! = 1 2! = 2!!! =2!

2! =22 = 1

__________________________________________________________________________________________________________________

𝑎!! =1𝑎! 2!! =

12! =

14

__________________________________________________________________________________________________________________

1𝑎!! = 𝑎!

12!! = 2! = 4

__________________________________________________________________________________________________________________

𝑎!𝑎! = 𝑎!!! 2!2! = 2!!! = 2!

__________________________________________________________________________________________________________________

𝑎!

𝑎! = 𝑎!!! 2!

2! = 2!!! = 2! = 2

__________________________________________________________________________________________________________________

𝑎𝑏

!=𝑎!

𝑏! 23

!

=2!

3! =49

__________________________________________________________________________________________________________________

𝑎𝑏

!!=𝑎!!

𝑏!! =𝑏!

𝑎! 12

!!

=1!!

2!! =2!

1 = 4

__________________________________________________________________________________________________________________

𝑎!!! = 𝑎

!! = 𝑎

!!

! 2!

!! = 2

!! = 2

!!

!

__________________________________________________________________________________________________________________

𝑎! ! = 𝑎!" = 𝑎!" = 𝑎! ! 2! ! = 2!∙! = 2 !∙! = 2! !

TrigonometryDoubleAngleFormulas

*Important

ThehalfangleanddoubleangleformulasalongwiththePythagoreanidentitiesareusedfrequentlythroughoutcalculus.Itisamustthatyoumemorizetheunderstandingandderivationsisfullycomprehended.

Derivationforsin 2𝜃 = 2 sin𝜃 cos𝜃:

sin 2𝜃 = sin 𝜃 + 𝜃 = sin𝜃 cos𝜃 + sin𝜃 cos𝜃 = 2 sin𝜃 cos𝜃

__________________________________________________________________________________________________________________

Derivationforcos 2𝜃 = 1− 2 sin! 𝜃:

cos(2𝜃) = cos! 𝜃 − sin! 𝜃 = 2 cos! 𝜃 − 1 = 1− 2 sin! 𝜃

__________________________________________________________________________________________________________________

Asonecansee,theseformulasareallderivedfromthePythagoreanidentitiesandtherearemanywaystofindthem.Ifthiscanbeunderstoodproperlythenmemorizingthemisnotentirelynecessary.

OtherDerivations:

cos2𝜃 = cos(𝜃 + 𝜃) = cos𝜃 cos𝜃 − sin𝜃 sin𝜃 = cos! 𝜃 − sin! 𝜃

_________________________________________________________________________________________________________________

_

cos 2𝜃 = cos(𝜃 + 𝜃) = cos𝜃 cos𝜃 − sin𝜃 sin𝜃 = cos! 𝜃 − sin! 𝜃 = cos! 𝜃 − (1− cos! 𝜃)

= cos!−1+ cos! 𝜃 = 2 cos! 𝜃 − 1

__________________________________________________________________________________________________________________

cos2𝜃 = cos(𝜃 + 𝜃) = cos𝜃 cos𝜃 − sin𝜃 sin𝜃 = cos! 𝜃 − sin! 𝜃

= 1− sin! 𝜃 − sin! 𝜃 = 1− 2 sin! 𝜃

__________________________________________________________________________________________________________________

tan 2𝜃 = tan 𝜃 + 𝜃 =tan𝜃 + tan𝜃1− tan𝜃 tan𝜃 =

2 tan𝜃1− tan! 𝜃

HalfAngleFormulas

sin! 𝜃 =12 1− cos 2𝜃

Derivation:

sin! 𝜃 = 1− cos! 𝜃 = 1− cos𝜃 cos𝜃 = 1−12 cos 𝜃 − 𝜃 + cos 𝜃 + 𝜃

= 1−12 cos 0 + cos 2𝜃 = 1−

12 1 + cos2𝜃 = 1−

12−

12 cos2𝜃

=12−

12 cos2𝜃 =

12 [1− cos(2𝜃)]

__________________________________________________________________________________________________________________

cos! 𝜃 =12 [1+ 𝑐𝑜𝑠 2𝜃 ]

Derivation:

cos! 𝜃 = 1− sin! 𝜃 = 1− sin𝜃 sin𝜃 = 1−12 cos(𝜃 − 𝜃 − cos 𝜃 + 𝜃 ]

= 1−12 cos0− cos 2𝜃 = 1−

12 1 − cos2𝜃 = 1−

12+

12 cos2𝜃

=12+

12 cos2𝜃 =

12 1+ cos 2𝜃

__________________________________________________________________________________________________________________

tan! 𝜃 =1− cos(2𝜃)1+ cos(2𝜃)

Derivation:

tan! 𝜃 = sec! 𝜃 − 1 =1

cos𝜃

!

− 1 =1

cos𝜃 cos𝜃 − 1 =1

12 cos 𝜃 − 𝜃 + cos 𝜃 + 𝜃

− 1

=2

1+ cos 2𝜃 − 1 =2

1+ cos2𝜃 −1+ cos 2𝜃1+ cos 2𝜃 =

2− 1+ cos2𝜃1+ cos 2𝜃

=1− cos 2𝜃1+ cos 2𝜃

SumandDifferenceFormulas

sin 𝛼 ± 𝛽 = sin𝛼 cos𝛽 ± cos𝛼 sin𝛽

__________________________________________________________________________________________________________________

cos(𝛼 ± 𝛽) = cos𝛼 cos𝛽 ∓ sin𝛼 cos𝛽_________________________________________________________________________________________________________________

_

tan 𝛼 ± 𝛽 =tan𝛼 ± tan𝛽1∓ tan𝛼 𝑡𝑎𝑛𝛽

Precalculus

EquationofaLine

𝑠𝑙𝑜𝑝𝑒 = 𝑚 =𝑦! − 𝑦!𝑥! − 𝑥!

𝑦 = 𝑚𝑥 + 𝑏

𝑦! − 𝑦! = 𝑚 𝑥! − 𝑥!

𝐴𝑥 + 𝐵𝑦 = 𝐶

EquationofParabola

Vertex: ℎ, 𝑘

𝑦 = 𝑎𝑥! + 𝑏𝑥 + 𝑐

𝑦 = 𝑎 𝑥 − ℎ ! + 𝑘

EquationofCircle

𝑥 − ℎ ! + 𝑦 − 𝑘 ! = 𝑟!

EquationofEllipse

RightPoint: ℎ + 𝑎, 𝑘

LeftPoint: ℎ − 𝑎, 𝑘

TopPoint: ℎ, 𝑘 + 𝑏

BottomPoint: ℎ, 𝑘 − 𝑏

𝑥 − ℎ !

𝑎! +𝑦 − 𝑘 !

𝑏! = 1

EquationofHyperbola

Center: ℎ, 𝑘 Slope:± !

!

Asymptotes:𝑦 = ± !!𝑥 − ℎ + 𝑘

Vertices: ℎ + 𝑎, 𝑘 , ℎ − 𝑎, 𝑘

𝑥 − ℎ !

𝑎! −𝑦 − 𝑘 !

𝑏! = 1

EquationofHyperbola

Center: ℎ, 𝑘 Slope:± !

!

Asymptotes:𝑦 = ± !!𝑥 − ℎ + 𝑘

Vertices: ℎ, 𝑘 + 𝑏 , ℎ, 𝑘 − 𝑏

𝑦 − 𝑘 !

𝑎! −𝑥 − ℎ !

𝑏! = 1

CalculusTangentline

Findtheequationofthetangentlineat𝑥 = 3 for 𝑦 = 𝑥!

Identify

𝑦 − 𝑓 𝑎 = 𝑓! 𝑎 𝑥 − 𝑎 , 𝑥! = 𝑎

𝑎 = 3

𝑓 𝑎 = 𝑓 3 = 3 ! = 9

𝑓! 𝑎 =𝑑𝑑𝑥 𝑥

! = 2𝑥

𝑓! 3 = 6Gobackandtakealookatthedifferencefromthelimitdefinitionprocessandthepowerruleprocess.

Nowplugeverythinginto𝑦 − 𝑦! = 𝑚 𝑥 − 𝑥!

𝑦 − 9 = 6 𝑥 − 3

∴ 𝑦 = 6𝑥 − 9

Graphingisalwaysgoodpractice

Implicitdifferentiation

Given𝑥𝑦 + 𝑦 = 𝑦! − 𝑥 find !"

!"

Simplytake !

!"ofthewholeequation

𝑑𝑑𝑥 𝑥𝑦 + 𝑦 = 𝑦! − 𝑥

⇒ 𝑑𝑑𝑥 𝑥𝑦 +

𝑑𝑑𝑥 𝑦 =

𝑑𝑑𝑥 𝑦

! −𝑑𝑑𝑥 𝑥

⇒ 𝑥𝑑𝑑𝑥 𝑦 + 𝑦

𝑑𝑑𝑥 𝑥 +

𝑑𝑦𝑑𝑥 = 2𝑦

𝑑𝑑𝑥 𝑦 − 1

⇒ 𝑥𝑑𝑦𝑑𝑥 + 𝑦 1 +

𝑑𝑦𝑑𝑥 = 2𝑦

𝑑𝑦𝑑𝑥 − 1

Feelfreetosubstitute𝑦! for !"

!"ifitistoomessy

⇒ 𝑥𝑦! + 𝑦 + 𝑦! = 2𝑦𝑦! − 1

⇒ 𝑥𝑦! + 𝑦! − 2𝑦𝑦! = −1− 𝑦

⇒ 𝑦! 𝑥 + 1− 2𝑦 = − 1+ 𝑦

⇒ 𝑦! =− 1+ 𝑦𝑥 + 1− 2𝑦 =

− 1+ 𝑦− 2𝑦 − 1− 𝑥 =

1+ 𝑦2𝑦 − 1− 𝑥

∴ 𝑑𝑦𝑑𝑥 =

𝑦 + 12𝑦 − 1− 𝑥

LinearAlgebraRankofmatrixandpivots

𝟏 1 , 𝑟𝑎𝑛𝑘 𝐴! = 1

𝟏 1 , 𝑟𝑎𝑛𝑘 𝐴! = 1

𝟏1 , 𝑟𝑎𝑛𝑘 𝐴! = 1

𝟏0 , 𝑟𝑎𝑛𝑘 𝐴! = 1

𝟏 1 1 , 𝑟𝑎𝑛𝑘 𝐴! = 1

𝟏 1 1 , 𝑟𝑎𝑛𝑘 𝐴!" = 1

𝟏11, 𝑟𝑎𝑛𝑘 𝐴! = 1

𝟏00, 𝑟𝑎𝑛𝑘 𝐴!! = 1

𝟏 00 𝟏 , 𝑟𝑎𝑛𝑘 𝐴! = 2

𝟏 1 11 1 11 1 1

, 𝑟𝑎𝑛𝑘 𝐴!" = 1

𝟏 0 00 𝟏 1 , 𝑟𝑎𝑛𝑘 𝐴! = 2

𝟏 1 11 1 −𝟏1 1 1

, 𝑟𝑎𝑛𝑘 𝐴!" = 2

𝟏 00 00 𝟏

, 𝑟𝑎𝑛𝑘 𝐴! = 2

𝟏 1 10 𝟏 10 0 𝟏

, 𝑟𝑎𝑛𝑘 𝐴!" = 3

Note:maxrankisthesmallerdimensionof𝑛×𝑚e.g.3×7 meansthat3isthehighestpossiblerank.Itgoeswiththetransposeaswelli.e.7×3stillhasahighestrankof3.

𝐴 = 1 2−1 −2

1 11 1

1 11 1 𝑅1+ 𝑅2 ⇐ 𝑅2

~ 𝟏 20 0

1 1𝟐 2

1 12 2 ⇒ 𝑟𝑎𝑛𝑘 𝐴 = 2

𝐴𝑥 = 𝑏 ⇒3 2 31 3 33 2 1

131~𝟏 0 00 𝟏 00 0 𝟏

−37870

, 𝑟𝑎𝑛𝑘 𝐴 = 3 𝑖. 𝑒. 𝐴 = 𝑓𝑢𝑙𝑙 𝑟𝑎𝑛𝑘

Lengthofavectorandtheunitvector

Givenavector𝒙 = 𝑥 = 𝑥!, 𝑥!, 𝑥!,… , 𝑥! =

𝑥!𝑥!𝑥!⋮𝑥!

Thelengthofthevectoristhemagnitudeofthevector

𝒙 = 𝑥!! + 𝑥!! + 𝑥!! +⋯+ 𝑥!!

Ex:

Findthelengthof 1,2,3,4

1,2,3,4 =

1234

⇒ 1,2,3,4 = 1! + 2! + 3! + 4! = 1+ 4+ 9+ 16 = 30 units

Example:

Fromthevectorabove,finditsunitvector.

𝑣𝒗 =

𝒗𝒗

⇒ 𝑣𝒗

= 𝒗𝒗 = 1 units

𝒙𝒙 =

11+ 4+ 9+ 16

1234

=1,2,3,430

=130,230,330,430

𝑥𝑥 =

130

!

+230

!

+330

!

+430

!

=130+

430+

930+

1630 =

3030 = 1 units

SolutionsofAugmentedMatrices

Considerthebasicscenarioi.e.rememberfromalgebrawhenyouhave𝑎𝑥 + 𝑏𝑦 = 𝑐and𝑑𝑥 +𝑒𝑦 = 𝑓?Rememberthatthesetwolineseitherlyeoneachother,intersectornevertouch,andthismeanstheyhaveeitherauniquesolution,infinitesolutions,onnosolution.Thesamegoeswith

𝑎𝑥 + 𝑏𝑦 + 𝑐𝑧 = 𝑑,exceptthisisaplane.

Forℝ!,considerthefollowingsystemanditsthreepossiblesolutionsafterreduction:

CoefficientMatrix

𝑎𝑥 + 𝑏𝑦 + 𝑐𝑧 = 𝑑𝑒𝑥 + 𝑓𝑦 + 𝑔𝑦 = ℎ𝑖𝑥 + 𝑗𝑦 + 𝑘𝑧 = 𝑙

⇒ 𝑎 𝑏 𝑐𝑒 𝑓 𝑔𝑖 𝑗 𝑘

𝑥𝑦𝑧=

𝑑ℎ𝑙

⇒ 𝑎 𝑏 𝑐𝑒 𝑓 𝑔𝑖 𝑗 𝑘

𝑑ℎ𝑙

TheCoefficientMatrix=𝑎 𝑏 𝑐𝑒 𝑓 𝑔𝑖 𝑗 𝑘

UniqueSolution

~1 0 00 1 00 0 1

∗∗∗⇒

𝑥𝑦𝑧=

∗∗∗

In2𝐷/3𝐷hereisasinglepointofintersection

InfiniteSolution

~1 0 00 1 00 0 0

∗∗0⇒

𝑥𝑦𝑧=

∗∗0+ 𝑠

001

In3𝐷twoplaneslieontopofeachotherIn2𝐷twolineslieontopofeachother

NoSolution

~1 0 00 1 00 0 0

∗∗∗⇒

𝑥𝑦0=

∗∗∗

Twoplanes/linesnevertouch

DifferentialEquations

First-OrderLinearNon-Homogeneous

cos! 𝑥 sin 𝑥𝑑𝑦𝑑𝑥 + cos

! 𝑥 𝑦 = 1

Form

𝑦! + 𝑃 𝑥 𝑦 = 𝑄 𝑥 ⇒ 𝑦 =1𝐼 𝑥 ∫ 𝑄 𝑥 𝐼 𝑥 𝑑𝑥 + 𝐶 ⇔ 𝐼 𝑥 = 𝑒 ! ! !"

cos! 𝑥 sin 𝑥𝑑𝑦𝑑𝑥 + cos

! 𝑥 𝑦 = 1

⇒ 1

cos! 𝑥 sin 𝑥 cos! 𝑥 sin 𝑥𝑑𝑦𝑑𝑥 + cos

! 𝑥 𝑦 = 1 ⇒ 𝑦! + cot 𝑥 𝑦 = sec! 𝑥 csc 𝑥

⇒ 𝑃 𝑥 = cot 𝑥 ∧ 𝑄 𝑥 = sec! 𝑥 csc 𝑥 ∧ 𝐼 𝑥 = 𝑒 !"# ! !" = sin 𝑥

⇒ 𝑦 =1

sin 𝑥 sec! 𝑥 csc 𝑥 sin 𝑥 𝑑𝑥 + 𝐶 = csc 𝑥 sec! 𝑥 𝑑𝑥 + 𝐶 = csc 𝑥 tan 𝑥 + 𝐶

⇒ 𝑦 = csc 𝑥 tan 𝑥 + 𝐶 csc 𝑥 =1

sin 𝑥sin 𝑥cos 𝑥 + 𝐶 csc 𝑥 = sec 𝑥 + 𝐶 csc 𝑥

∴ cos! 𝑥 sin 𝑥𝑑𝑦𝑑𝑥 + cos

! 𝑥 𝑦 = 1 ⇔ 𝑦 = sec 𝑥 + 𝐶 csc 𝑥

OrderandLinearity

𝑦! + 𝑥𝑦!! − !!!

!!!= sin 𝑥𝑦 Sixth-Order-NonlinearandNonhomogeneous

𝑥𝑦!! − !!!

!!!= sin 𝑥 Sixth-Order-LinearandNonhomogeneous

𝑦′′+ 𝑦′+ 𝑦𝑥 = 0Second-Order-LinearandHomogeneous

𝑦′′+ 𝑦𝑦′ = 0Second-Order-NonlinearandHomogeneousNote:Althoughthepowerofyis1inthiscase,itisdependentupony’makingitnonlinear.

𝑦′′′+ 𝑦! + 𝑥𝑒! = 0Third-Order-NonlinearandHomogeneousReductionofOrderProcess:

GivenasecondorderlinearhomogeneousDEoftheform𝑦!! + 𝑃 𝑥 𝑦! + 𝑄 𝑥 = 0accompanied

with𝑦!(𝑥)

Solution

Sincethefirstsolutionisgiven,youmustfindthesecondsolution,whichis:

𝑦! 𝑥 = 𝑦! 𝑥𝑒! ! ! !"

𝑦! 𝑥 ! 𝑑𝑥, ∴ 𝑦 = 𝑐!𝑦! + 𝑐! 𝑦! 𝑥𝑒! ! ! !"

𝑦! 𝑥 ! 𝑑𝑥

Example:

𝑥!𝑦!! + 2𝑥𝑦! − 6𝑦 = 0, 𝑦! = 𝑥!

Find𝑃 𝑥

1𝑥! 𝑥!𝑦!! + 2𝑥𝑦! − 6𝑦 = 0 ⇒ 𝑦!! +

2𝑥 𝑦

! −6𝑥! 𝑦 = 0 ⇒ 𝑃 𝑥 =

2𝑥

∴ 𝑦! = 𝑥!𝑒!

!!!"

𝑥! ! 𝑑𝑥 = 𝑥!𝑒!! !" !

𝑥! 𝑑𝑥 = 𝑥!𝑒!" !!!

𝑥! 𝑑𝑥 = 𝑥!𝑥!!

𝑥! 𝑑𝑥 = 𝑥! 𝑥!! 𝑑𝑥

= 𝑥!1−5 𝑥

!! = −15 𝑥

!! ⇒ 𝑦! =1𝑥!

Theconstantcanbeignoredbecauseaconstanttimesaconstantisaconstant

∴ 𝑦 = 𝑐!𝑥! +

𝑐!𝑥!

Atthispointitshouldbecomeobviousthat𝑐! + 𝑐! +⋯+ 𝑐! = 𝐶,thisisalsotruefornumbersi.e.𝑐! + 5+ 𝑒 + ln 10 + 𝑒!! + 6𝑐! = 𝐶.Inotherwords:aconstantwithaconstantisaconstant.

PhysicsVectors Notation

𝑎 = 𝑎!,𝑎! in2Dor𝑎 = 𝑎!,𝑎!,𝑎! in3D

𝑎 ± 𝑏 = 𝑎!,𝑎! ± 𝑏!, 𝑏! = 𝑎! ± 𝑏!,𝑎! ± 𝑏!

𝑎 ± 𝑏 = 𝑎!,𝑎!,𝑎! ± 𝑏!, 𝑏!, 𝑏! = 𝑎! ± 𝑏!,𝑎! ± 𝑏!,𝑎! ± 𝑏! Visually

DotProduct

𝑎 ⋅ 𝑏 = 𝑎!,𝑎! ⋅ 𝑏!, 𝑏! = 𝑎!𝑏! + 𝑎!𝑏!

𝑎 ⋅ 𝑏 = 𝑎!,𝑎!,𝑎! ⋅ 𝑏!, 𝑏!, 𝑏! = 𝑎!𝑏! + 𝑎!𝑏! + 𝑎!𝑏!CrossProduct

𝑎×𝑏 = −𝑏×𝑎

𝑎×𝑏 =𝚤 𝚥 𝑘𝑎! 𝑎! 𝑎!𝑏! 𝑏! 𝑏!

= 𝚤

𝑎! 𝑎!𝑏! 𝑏! − 𝚥

𝑎! 𝑎!𝑏! 𝑏! + 𝑘

𝑎! 𝑎!𝑏! 𝑏!

= 𝚤 𝑎! 𝑏! − 𝑎! 𝑏! − 𝚥 𝑎! 𝑏! − 𝑎! 𝑏! + 𝑘 𝑎! 𝑏! − 𝑎! 𝑏!

𝚤, 𝚥,and𝑘arecalledunitvectors.Aunitvector,isavectoroflength1

𝚤 = 1, 0, 0 , 𝚥 = 0, 1, 0 , 𝑘 = 0, 0, 1

= 𝑎! 𝑏! − 𝑎! 𝑏! , 0, 0 − 0, 𝑎! 𝑏! − 𝑎! 𝑏! , 0 + 0, 0, 𝑎! 𝑏! − 𝑎! 𝑏!

= 𝑎! 𝑏! − 𝑎! 𝑏! , 𝑎! 𝑏! − 𝑎! 𝑏! , 𝑎! 𝑏! − 𝑎! 𝑏!

MagnitudeorLengthofavectorAboldletterisavectori.e.𝑎 = 𝒂 = 𝑎!,𝑎!,𝑎!

2𝐷, 𝒂 = 𝑎 = 𝑎 = 𝑎!! + 𝑎!!

3𝐷, 𝒂 = 𝑎 = 𝑎 = 𝑎!! + 𝑎!! + 𝑎!!

UnitizingavectorTomakethevectorbeoflength1butpreservethedirection.

2𝐷, 𝑎 =𝑎𝑎 =

𝑎!,𝑎!𝑎!! + 𝑎!!

3𝐷, 𝑎 =𝑎𝑎 =

𝑎!,𝑎!,𝑎!𝑎!! + 𝑎!! + 𝑎!!

ResultantVector

𝑅 = 𝑎 + 𝑏Inphysicsyouwillbeusuallybegiventhevectore.g.(e.g.=forexample)𝑣(𝑣=velocity)

Theresultantvector,𝑣wouldbeavectorthatcanbebrokenintoa𝑥and𝑦component.Anglewithrespecttox-axis Anglewithrespecttoy-axis

𝑣 = 𝑣 cos𝜃 , 𝑣 sin𝜃 𝑣 = 𝑣 sin𝜙 , 𝑣 cos𝜙

𝑣,𝑣! = 𝑣 cos𝜃 , 𝑣! = 𝑣 cos𝜃 , 0𝑣! = 𝑣 sin𝜃 , 𝑣! = 0, 𝑣 sin𝜃

𝑣,𝑣! = 𝑣 sin𝜙 , 𝑣! = 𝑣 sin𝜙 , 0𝑣! = 𝑣 cos𝜙 , 𝑣! = 0, 𝑣 cos𝜙

𝑅 = 𝑣 = 𝑣! + 𝑣! = 𝑣 cos𝜃 , 0 + 0, 𝑣 sin𝜃 = 𝑣 cos𝜃 , 𝑣 sin𝜃 𝑣 = 𝑣 cos𝜃 ! + 𝑣 sin𝜃 ! = 𝑣! cos! 𝜃 + 𝑣! sin! 𝜃 = 𝑣! cos! 𝜃 + sin! 𝜃 = 𝑣! 1 = 𝑣

Thismaybeslightlyconfusingwiththenotationbecauseofthevectorsbutinphysics,youwillbegivenanumberforthevectori.e.𝑣 = −25!

!,𝜃 = 25°(avectorhasmagnitudeanddirection,

whichmeansitcanbe𝑣 = −25!!cos 25° , −25!

!sin 25° orformagnitude 𝑣 = 𝑣 = 25!

!.

QuickReferenceArithmetic

𝑎𝑏 ± 𝑎𝑐 = 𝑎 𝑏 ± 𝑐 = 𝑏 ± 𝑐 𝑎 𝑎𝑏𝑐 =

𝑎𝑏𝑐

𝑎𝑏 ±

𝑐𝑑 =

𝑎𝑑 ± 𝑏𝑐𝑏𝑑

𝑎 − 𝑏𝑐 − 𝑑 =

𝑏 − 𝑎𝑑 − 𝑐

𝑎𝑏 + 𝑎𝑐𝑎 = 𝑏 + 𝑐,𝑎 ≠ 0 𝑎

𝑏𝑐 =

𝑎𝑏𝑐

𝑎𝑏𝑐=

𝑎1 ∙

𝑐𝑏 =

𝑎𝑐𝑏

𝑎 ± 𝑏𝑐 =

𝑎𝑐 ±

𝑏𝑐

𝑎𝑏𝑐𝑑

=𝑎𝑏 ∙𝑑𝑐 =

𝑎𝑑𝑏𝑐

Exponential

𝑎! = 𝑎 𝑎! = 1 𝑎!! =1𝑎!

1𝑎!! = 𝑎! 𝑎!𝑎! = 𝑎!!!

𝑎!

𝑎! = 𝑎!!! 𝑎𝑏

!=𝑎!

𝑏! 𝑎𝑏

!!=𝑏!

𝑎! 𝑎!!! = 𝑎

!!

! 𝑎! ! = 𝑎! !

𝑎!!= 𝑎!" = 𝑎

!!" 𝑎!! = 𝑎,𝑛 𝑖𝑠 𝑜𝑑𝑑 𝑎!! = 𝑎 ,𝑛 𝑖𝑠 𝑒𝑣𝑒𝑛

𝑎 = 𝑎! = 𝑎!! = 𝑎!! 𝑎!! = 𝑎

!!

𝑎𝑏

!=

𝑎!

𝑏! =𝑎!!

𝑏!!=

𝑎𝑏

!!

Fractions

𝑎𝑏 ±

𝑐𝑑 =

𝑎𝑑 ± 𝑏𝑐𝑏𝑑

𝑔 𝑥𝑓 𝑥 ±

ℎ 𝑥𝑟 𝑥 =

𝑔 𝑥 𝑟 𝑥 ± 𝑓 𝑥 ℎ 𝑥𝑓 𝑥 𝑟 𝑥

Logarithmicln 𝑏ln 𝑎 = log! 𝑏 𝑦 = log! 𝑥⇔ 𝑥 = 𝑏! 𝑒 ≈ 2.72 log! 𝑎 = 1

log! 1 = 0 log! 𝑎! = 𝑢 log! 𝑢 = ln𝑢 log! 𝑢! = 𝑏 log! 𝑢

log! 𝑢𝑣 = log! 𝑢 + log! 𝑣 log!𝑢𝑣 = log! 𝑢 − log! 𝑣 log! 𝑏 =

ln 𝑏ln 𝑎

𝑣 = ln𝑢 ⇒ 𝑢 = 𝑒! 𝑣 = 𝑒! ⇒ 𝑢 = ln 𝑣 𝑒 =1𝑛!

!

!!!

ln𝑎 = undefined,𝑎 ≤ 0 ln 1 = 0 ln 𝑒! = 𝑢 ⇒ 𝑒!"! = 𝑢ln 𝑒! = 1 ⇒ 𝑒!" ! = 1 ln𝑢! = 𝑏 ln𝑢 ln𝑢𝑣 = ln𝑢 + ln 𝑣 ln

𝑢𝑣 = ln𝑢 − ln 𝑣