The Ultimate Cheat Sheet for Math & Physics - WeSolveThem.com

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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 Copyright © WeSolveThem LLC

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Quadratic Formula .................................................................................................................................................................................. 20

About: This book covers all the formula, equations tips and tricks an undergraduate STEM major requires for Algebra, Trigonometry, Precalculus, Calculus (all levels/areas), Linear Algebra, Differential Equations, and Physics (Mechanics, E&M, Optics…) The book is applicable to any STEM student at any point of their career. It can act as a review, a guided assistant and or tool for students outside of college. Arithmetic

Algebra

𝑎𝑏 ± 𝑎𝑐 = 𝑎 𝑏 ± 𝑐 = 𝑏 ± 𝑐 𝑎 10 ± 6 = 2 ∙ 5 ± 2 ∙ 3 = 2 5 ± 3 = 5 ± 3 2 _________________________________________________________________________________________________________________ _ 𝑎 1 1 𝑎 𝑏 = 2 = 2 = 1 ∙ 1 = 1 = 1 3 2 3 2∙3 6 𝑐 𝑏𝑐 3 1 _________________________________________________________________________________________________________________ _ 𝑎 𝑐 𝑎𝑑 ± 𝑏𝑐 1 3 1∙4±2∙3 4±6 ± = ± = = 𝑏 𝑑 𝑏𝑑 2 4 2∙4 8 _________________________________________________________________________________________________________________ _ 𝑎−𝑏 𝑏−𝑎 1 − 2 −(−1 + 2) 2 − 1 = = = 𝑐−𝑑 𝑑−𝑐 3 − 4 −(−3 + 4) 4 − 3 _________________________________________________________________________________________________________________ _ 𝑎𝑏 + 𝑎𝑐 12 ± 16 12 16 = 𝑏 + 𝑐, 𝑎 ≠ 0 = ± =3±4 𝑎 4 4 4 _________________________________________________________________________________________________________________ _ 𝑏 𝑎𝑏 16 4 ∙ 4 4 𝑎 = = =4 𝑐 𝑐 5 5 5 _________________________________________________________________________________________________________________ _ Copyright © WeSolveThem LLC 2

2 𝑎 𝑎 𝑐 𝑎𝑐 2 2 4 8 = ∙ = = 1 = ∙ = 3 3 𝑏 1 𝑏 𝑏 1 3 3 4 4 𝑐 _________________________________________________________________________________________________________________ _ 𝑎±𝑏 𝑎 𝑏 12 ± 16 12 16 = ± = ± 𝑐 𝑐 𝑐 5 5 5 _________________________________________________________________________________________________________________ _ 𝑎 1 𝑏 = 𝑎 ∙ 𝑑 = 𝑎𝑑 2 = 1 ∙ 4 = 4 = 2 𝑐 3 𝑏 𝑐 𝑏𝑐 2 3 6 3 𝑑 4 _________________________________________________________________________________________________________________ _ 𝑖𝑓 𝑎 ± 𝑏 = 0 𝑡ℎ𝑒𝑛 𝑎 = ∓𝑏 𝑥 ± 2 = 0 ⇒ 𝑥 = ∓2 Exponents ! 𝑎 =𝑎 2 = 2! _________________________________________________________________________________________________________________ _ 2! 2 𝑎! = 1 2! = 2!!! = ! = = 1 2 2 _________________________________________________________________________________________________________________ _ 1 1 1 𝑎!! = ! 2!! = ! = 𝑎 2 4 _________________________________________________________________________________________________________________ _ 1 1 ! = 𝑎 = 2! = 4 𝑎!! 2!! _________________________________________________________________________________________________________________ _ ! ! !!! 𝑎 𝑎 =𝑎 2! 2! = 2!!! = 2! _________________________________________________________________________________________________________________ _ Copyright © WeSolveThem LLC

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!

𝑎 2! !!! = 𝑎 = 2!!! = 2! = 2 𝑎! 2! _________________________________________________________________________________________________________________ _ ! ! 𝑎 𝑎 2 ! 2! 4 = ! = ! = 𝑏 𝑏 3 3 9 _________________________________________________________________________________________________________________ _ !! ! !! 𝑎 𝑎 𝑏 1 !! 1!! 2! = !! = ! = !! = = 4 𝑏 𝑏 𝑎 2 2 1 _________________________________________________________________________________________________________________ _ 𝑎!

! !

!

! !

2!

= 𝑎! = 𝑎!

! !

!

! !

= 2! = 2!

_________________________________________________________________________________________________________________ _ 𝑎! ! = 𝑎!" = 𝑎!" = 𝑎! ! 2! ! = 2!∙! = 2 !∙! = 2! !

Trigonometry

Double Angle Formulas *Important The half angle and double angle formulas along with the Pythagorean identities are used frequently throughout calculus. It is a must that you memorize the understanding and derivations is fully comprehended.

For a detailed list of all identities, see the reference sheets in the back of the book. Derivation for sin 2𝜃 = 2 sin 𝜃 cos 𝜃: sin 2𝜃 = sin 𝜃 + 𝜃 = sin 𝜃 cos 𝜃 + sin 𝜃 cos 𝜃 = 2 sin 𝜃 cos 𝜃 _________________________________________________________________________________________________________________ _ Derivation for cos 2𝜃 = 1 − 2 sin! 𝜃: cos(2𝜃) = cos ! 𝜃 − sin! 𝜃 = 2 cos ! 𝜃 − 1 = 1 − 2 sin! 𝜃

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_________________________________________________________________________________________________________________ _ As one can see, these formulas are all derived from the Pythagorean identities and there are many ways to find them. If this can be understood properly then memorizing them is not entirely necessary.

Other Derivations: cos 2𝜃 = cos(𝜃 + 𝜃) = cos 𝜃 cos 𝜃 − sin 𝜃 sin 𝜃 = cos ! 𝜃 − sin! 𝜃 _________________________________________________________________________________________________________________ _ cos 2𝜃 = cos(𝜃 + 𝜃) = cos 𝜃 cos 𝜃 − sin 𝜃 sin 𝜃 = cos ! 𝜃 − sin! 𝜃 = cos ! 𝜃 − (1 − cos ! 𝜃) = cos ! −1 + cos ! 𝜃 = 2 cos ! 𝜃 − 1 _________________________________________________________________________________________________________________ _ cos 2𝜃 = cos(𝜃 + 𝜃) = cos 𝜃 cos 𝜃 − sin 𝜃 sin 𝜃 = cos ! 𝜃 − sin! 𝜃 = 1 − sin! 𝜃 − sin! 𝜃 = 1 − 2 sin! 𝜃 _________________________________________________________________________________________________________________ _ tan 𝜃 + tan 𝜃 2 tan 𝜃 tan 2𝜃 = tan 𝜃 + 𝜃 = = 1 − tan 𝜃 tan 𝜃 1 − tan! 𝜃 Half Angle Formulas 1 sin! 𝜃 = 1 − cos 2𝜃 2 Derivation: 1 sin! 𝜃 = 1 − cos ! 𝜃 = 1 − cos 𝜃 cos 𝜃 = 1 − cos 𝜃 − 𝜃 + cos 𝜃 + 𝜃 2 1 1 1 1 = 1 − cos 0 + cos 2𝜃 = 1 − 1 + cos 2𝜃 = 1 − − cos 2𝜃 2 2 2 2 1 1 1 = − cos 2𝜃 = [1 − cos(2𝜃)] 2 2 2 Copyright © WeSolveThem LLC

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_________________________________________________________________________________________________________________ _ 1 cos ! 𝜃 = [1 + 𝑐𝑜𝑠 2𝜃 ] 2 Derivation: 1 cos ! 𝜃 = 1 − sin! 𝜃 = 1 − sin 𝜃 sin 𝜃 = 1 − cos(𝜃 − 𝜃 − cos 𝜃 + 𝜃 ] 2 1 1 1 1 = 1 − cos 0 − cos 2𝜃 = 1 − 1 − cos 2𝜃 = 1 − + cos 2𝜃 2 2 2 2 1 1 1 = + cos 2𝜃 = 1 + cos 2𝜃 2 2 2 _________________________________________________________________________________________________________________ _ 1 − cos(2𝜃) tan! 𝜃 = 1 + cos(2𝜃) Derivation: ! 1 1 1 tan! 𝜃 = sec ! 𝜃 − 1 = −1= −1= − 1 1 cos 𝜃 cos 𝜃 cos 𝜃 cos 𝜃 − 𝜃 + cos 𝜃 + 𝜃 2 2 2 1 + cos 2𝜃 2 − 1 + cos 2𝜃 = −1= − = 1 + cos 2𝜃 1 + cos 2𝜃 1 + cos 2𝜃 1 + cos 2𝜃 1 − cos 2𝜃 = 1 + cos 2𝜃 Sum and Difference Formulas sin 𝛼 ± 𝛽 = sin 𝛼 cos 𝛽 ± cos 𝛼 sin 𝛽 _________________________________________________________________________________________________________________ _ cos(𝛼 ± 𝛽) = cos 𝛼 cos 𝛽 ∓ sin 𝛼 cos 𝛽 _________________________________________________________________________________________________________________ _ tan 𝛼 ± tan 𝛽 tan 𝛼 ± 𝛽 = 1 ∓ tan 𝛼 𝑡𝑎𝑛𝛽 Copyright © WeSolveThem LLC

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Precalculus



Equation of a Line 𝑠𝑙𝑜𝑝𝑒 = 𝑚 =

𝑦! − 𝑦! 𝑥! − 𝑥!

Equation of Parabola Vertex: ℎ, 𝑘 Equation of Circle

Center: ℎ, 𝑘 Radius: 𝑟 Equation of Ellipse

𝑦 = 𝑚𝑥 + 𝑏 𝑦! − 𝑦! = 𝑚 𝑥! − 𝑥! 𝐴𝑥 + 𝐵𝑦 = 𝐶 𝑦 = 𝑎𝑥 ! + 𝑏𝑥 + 𝑐 𝑦 = 𝑎 𝑥 − ℎ ! + 𝑘 ! 𝑥 − ℎ + 𝑦 − 𝑘 ! = 𝑟!

𝑥−ℎ 𝑎!

!

+

Right Point: ℎ + 𝑎, 𝑘

𝑦−𝑘 𝑏!

!

= 1

Left Point: ℎ − 𝑎, 𝑘 Top Point: ℎ, 𝑘 + 𝑏

Bottom Point: ℎ, 𝑘 − 𝑏 Equation of Hyperbola Center: ℎ, 𝑘 ! Slope: ± !

𝑥−ℎ 𝑎!

!

𝑦−𝑘 − 𝑏!

!

= 1

!

Asymptotes: 𝑦 = ± ! 𝑥 − ℎ + 𝑘 Vertices: ℎ + 𝑎, 𝑘 , ℎ − 𝑎, 𝑘 Equation of Hyperbola Center: ℎ, 𝑘 ! Slope: ± !

! 𝑦−𝑘 𝑥−ℎ − 𝑎! 𝑏!

!

= 1

!



Asymptotes: 𝑦 = ± ! 𝑥 − ℎ + 𝑘 Vertices: ℎ, 𝑘 + 𝑏 , ℎ, 𝑘 − 𝑏

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Calculus Tangent line Find the equation of the tangent line at 𝑥 = 3 for 𝑦 = 𝑥 ! Identify 𝑦 − 𝑓 𝑎 = 𝑓! 𝑎 𝑥 − 𝑎 , 𝑥! = 𝑎 𝑎 = 3 𝑓 𝑎 = 𝑓 3 = 3 ! = 9 𝑑 ! 𝑓! 𝑎 = 𝑥 = 2𝑥 𝑑𝑥 𝑓 ! 3 = 6 Go back and take a look at the difference from the limit definition process and the power rule process.

Now plug everything into 𝑦 − 𝑦! = 𝑚 𝑥 − 𝑥! 𝑦−9=6 𝑥−3 ∴ 𝑦 = 6𝑥 − 9 Graphing is always good practice



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Implicit differentiation Given 𝑥𝑦 + 𝑦 = 𝑦 ! − 𝑥 find

!" !"



!

Simply take !" of the whole equation



𝑑 𝑥𝑦 + 𝑦 = 𝑦 ! − 𝑥 𝑑𝑥

𝑑 𝑑 𝑑 ! 𝑑 ⇒ 𝑥𝑦 + 𝑦= 𝑦 − 𝑥 𝑑𝑥 𝑑𝑥 𝑑𝑥 𝑑𝑥 𝑑 𝑑 𝑑𝑦 𝑑 ⇒ 𝑥 𝑦+𝑦 𝑥 + = 2𝑦 𝑦 − 1 𝑑𝑥 𝑑𝑥 𝑑𝑥 𝑑𝑥 𝑑𝑦 𝑑𝑦 𝑑𝑦 ⇒ 𝑥 +𝑦 1 + = 2𝑦 − 1 𝑑𝑥 𝑑𝑥 𝑑𝑥 !" ! Feel free to substitute 𝑦 for !" if it is too messy ! ! ! ⇒ 𝑥𝑦 + 𝑦 + 𝑦 = 2𝑦𝑦 − 1 ! ! ! ⇒ 𝑥𝑦 + 𝑦 − 2𝑦𝑦 = −1 − 𝑦 ! ⇒ 𝑦 𝑥 + 1 − 2𝑦 = − 1 + 𝑦 − 1+𝑦 − 1+𝑦 1+𝑦 ⇒ 𝑦! = = = 𝑥 + 1 − 2𝑦 − 2𝑦 − 1 − 𝑥 2𝑦 − 1 − 𝑥 𝑑𝑦 𝑦+1 ∴ = 𝑑𝑥 2𝑦 − 1 − 𝑥



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Linear Algebra Rank of matrix and pivots

1,

𝟏

𝟏 , 1

𝟏 0

𝟏 0 0

𝟏 , 0

𝑟𝑎𝑛𝑘 𝐴! = 1

𝟏

0 , 1

0 0 , 𝟏



𝟏 1 1

𝑟𝑎𝑛𝑘 𝐴! = 2

𝟏 1 1

𝑟𝑎𝑛𝑘 𝐴! = 2

𝟏 0 0

1 1,

𝑟𝑎𝑛𝑘 𝐴! = 2

𝑟𝑎𝑛𝑘 𝐴!" = 1 𝑟𝑎𝑛𝑘 𝐴!! = 1

1 1 1 1 , 1 1 1 1 1 −𝟏 , 1 1 1 𝟏 0

𝑟𝑎𝑛𝑘 𝐴! = 1 𝑟𝑎𝑛𝑘 𝐴! = 1

𝟏 0 , 0

𝑟𝑎𝑛𝑘 𝐴! = 1

0 , 𝟏

0 𝟏



1,

𝟏



𝟏 1 , 1 𝟏 0

𝑟𝑎𝑛𝑘 𝐴! = 1 𝑟𝑎𝑛𝑘 𝐴! = 1

1 1,

𝟏



1 1 , 𝟏

𝑟𝑎𝑛𝑘 𝐴!" = 1 𝑟𝑎𝑛𝑘 𝐴!" = 2 𝑟𝑎𝑛𝑘 𝐴!" = 3

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Note: max rank is the smaller dimension of 𝑛×𝑚 e.g. 3×7 means that 3 is the highest possible rank. It goes with the transpose as well i.e. 7×3 still has a highest rank of 3. 1 2 1 1 1 1 𝑅1 + 𝑅2 ⇐ 𝑅2 𝟏 2 1 1 1 1 𝐴= ⇒ 𝑟𝑎𝑛𝑘 𝐴 = 2 −1 −2 1 1 1 1 ~ 0 0 𝟐 2 2 2 3 3 2 31 𝟏 0 0 −7 𝐴𝑥 = 𝑏 ⇒ 1 3 3 3 ~ 0 𝟏 0 8 , 𝑟𝑎𝑛𝑘 𝐴 = 3 𝑖. 𝑒. 𝐴 = 𝑓𝑢𝑙𝑙 𝑟𝑎𝑛𝑘 3 2 11 0 0 𝟏 7 0



Length of a vector and the unit vector



Given a vector 𝒙 = 𝑥 = 𝑥! , 𝑥! , 𝑥! , … , 𝑥!

𝑥! 𝑥! = 𝑥! ⋮ 𝑥!

The length of the vector is the magnitude of the vector 𝒙 =

1 2 1,2,3,4 = ⇒ 3 4

𝑥!! + 𝑥!! + 𝑥!! + ⋯ + 𝑥!!

Ex: Find the length of 1,2,3,4 1,2,3,4

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=

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

11

Example: From the vector above, find its unit vector. 𝑣 𝒗 𝑣 𝒗 = ⇒ = = 1 units 𝒗 𝒗 𝒗 𝒗 1 𝒙 1 1,2,3,4 1 2 3 4 2 = = = , , , 𝒙 1 + 4 + 9 + 16 3 30 30 30 30 30 4 𝑥 𝑥

=

1 30

!

+

2

!

30

+

3 30

!

+

4 30

!

=

1 4 9 16 + + + = 30 30 30 30

30 = 1 units 30



Solutions of Augmented Matrices



Consider the basic scenario i.e. remember from algebra when you have 𝑎𝑥 + 𝑏𝑦 = 𝑐 and 𝑑𝑥 + 𝑒𝑦 = 𝑓? Remember that these two lines either lye on each other, intersect or never touch, and this means they have either a unique solution, infinite solutions, on no solution. The same goes with 𝑎𝑥 + 𝑏𝑦 + 𝑐𝑧 = 𝑑, except this is a plane. ! For ℝ , consider the following system and its three possible solutions after reduction: Coefficient Matrix 𝑎𝑥 + 𝑏𝑦 + 𝑐𝑧 = 𝑑 𝑎 𝑏 𝑐 𝑥 𝑎 𝑏 𝑐 𝑑 𝑑 𝑒 𝑓 𝑔 𝑒 𝑓 𝑔ℎ 𝑒𝑥 + 𝑓𝑦 + 𝑔𝑦 = ℎ ⇒ 𝑦 = ℎ ⇒ 𝑖 𝑗 𝑘 𝑧 𝑖 𝑗 𝑘 𝑙 𝑖𝑥 + 𝑗𝑦 + 𝑘𝑧 = 𝑙 𝑙 𝑎 𝑏 𝑐 The Coefficient Matrix = 𝑒 𝑓 𝑔 𝑖 𝑗 𝑘

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Unique Solution 𝑥 ∗ 1 0 0∗ ~ 0 1 0∗ ⇒ 𝑦 = ∗ 𝑧 ∗ 0 0 1∗ In 2𝐷/3𝐷 here is a single point of intersection

Infinite Solution 𝑥 ∗ 1 0 0∗ 0 ~ 0 1 0 ∗ ⇒ 𝑦 = ∗ +𝑠 0 𝑧 0 0 0 00 1 In 3𝐷 two planes lie on top of each other In 2𝐷 two lines lie on top of each other



No Solution

𝑥 ∗ 1 0 0∗ ~ 0 1 0∗ ⇒ 𝑦 = ∗ ∗ 0 0 0 0∗ Two planes/lines never touch









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Differential Equations First-Order Linear Non-Homogeneous

𝑑𝑦 cos ! 𝑥 sin 𝑥 + cos ! 𝑥 𝑦 = 1 𝑑𝑥 Form 1 𝑦! + 𝑃 𝑥 𝑦 = 𝑄 𝑥 ⇒ 𝑦= ∫ 𝑄 𝑥 𝐼 𝑥 𝑑𝑥 + 𝐶 ⇔ 𝐼 𝑥 = 𝑒 ! ! !" 𝐼 𝑥 𝑑𝑦 cos ! 𝑥 sin 𝑥 + cos ! 𝑥 𝑦 = 1 𝑑𝑥 1 𝑑𝑦 ⇒ cos ! 𝑥 sin 𝑥 + cos ! 𝑥 𝑦 = 1 ⇒ 𝑦 ! + cot 𝑥 𝑦 = sec ! 𝑥 csc 𝑥 cos ! 𝑥 sin 𝑥 𝑑𝑥 ! ⇒ 𝑃 𝑥 = cot 𝑥 ∧ 𝑄 𝑥 = sec 𝑥 csc 𝑥 ∧ 𝐼 𝑥 = 𝑒 !"# ! !" = sin 𝑥 1 ⇒ 𝑦= sec ! 𝑥 csc 𝑥 sin 𝑥 𝑑𝑥 + 𝐶 = csc 𝑥 sec ! 𝑥 𝑑𝑥 + 𝐶 = csc 𝑥 tan 𝑥 + 𝐶 sin 𝑥 1 sin 𝑥 ⇒ 𝑦 = csc 𝑥 tan 𝑥 + 𝐶 csc 𝑥 = + 𝐶 csc 𝑥 = sec 𝑥 + 𝐶 csc 𝑥 sin 𝑥 cos 𝑥 𝑑𝑦 ∴ cos ! 𝑥 sin 𝑥 + cos ! 𝑥 𝑦 = 1 ⇔ 𝑦 = sec 𝑥 + 𝐶 csc 𝑥 𝑑𝑥 Order and Linearity !!!

𝑦 ! + 𝑥𝑦 !! − !! ! = sin 𝑥𝑦 Sixth-Order-Nonlinear and Nonhomogeneous !!!

𝑥𝑦 !! − !! ! = sin 𝑥 Sixth-Order-Linear and Nonhomogeneous 𝑦′′ + 𝑦′ + 𝑦𝑥 = 0 Second-Order-Linear and Homogeneous Copyright © WeSolveThem LLC

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𝑦′′ + 𝑦𝑦′ = 0 Second-Order-Nonlinear and Homogeneous Note: Although the power of y is 1 in this case, it is dependent upon y’ making it nonlinear. 𝑦′′′ + 𝑦 ! + 𝑥𝑒 ! = 0 Third-Order-Nonlinear and Homogeneous Reduction of Order Process: Given a second order linear homogeneous DE of the form 𝑦 !! + 𝑃 𝑥 𝑦 ! + 𝑄 𝑥 = 0 accompanied with 𝑦! (𝑥) Solution Since the first solution is given, you must find the second solution, which is: ! ! ! !" 𝑒 𝑒 ! ! ! !" 𝑦! 𝑥 = 𝑦! 𝑥 𝑑𝑥, ∴ 𝑦 = 𝑐 𝑦 + 𝑐 𝑦 𝑥 𝑑𝑥 ! ! ! ! 𝑦! 𝑥 ! 𝑦! 𝑥 ! Example:

𝑥 𝑦 + 2𝑥𝑦 − 6𝑦 = 0, 𝑦! = 𝑥 ! Find 𝑃 𝑥 1 ! !! 2 ! 6 ! !! 𝑥 𝑦 + 2𝑥𝑦 − 6𝑦 = 0 ⇒ 𝑦 + 𝑦 − !𝑦 = 0 𝑥! 𝑥 𝑥 ! !!

!

⇒

2 𝑃 𝑥 = 𝑥

!!

𝑒 !! !" ! 𝑒 !" ! 𝑥 !! ! ! ∴ 𝑦! = 𝑥 𝑑𝑥 = 𝑥 𝑑𝑥 = 𝑥 𝑑𝑥 = 𝑥 ! 𝑥 !! 𝑑𝑥 𝑥! 𝑥! 𝑥! 1 !! 1 1 = 𝑥! 𝑥 = − 𝑥 !! ⇒ 𝑦! = ! −5 5 𝑥 The constant can be ignored because a constant times a constant is a constant 𝑐! ∴ 𝑦 = 𝑐! 𝑥 ! + ! 𝑥 At this point it should become obvious that 𝑐! + 𝑐! + ⋯ + 𝑐! = 𝐶, this is also true for numbers i.e. 𝑐! + 5 + 𝑒 + ln 10 + 𝑒 !! + 6𝑐! = 𝐶. In other words: a constant with a constant is a constant. !



𝑒 ! !!" 𝑑𝑥 = 𝑥 ! 𝑥! !

!

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Physics Vectors Notation

𝑎 = 𝑎! , 𝑎! in 2D or 𝑎 = 𝑎! , 𝑎! , 𝑎! in 3D

Addition/Subtraction Visually

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



Dot Product Cross Product

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

𝑎×𝑏 = −𝑏×𝑎

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𝑎! =𝚤 𝑏 !



= 𝚤 𝑎! 𝑏! − 𝑎! 𝑏!



𝚤 𝑎×𝑏 = 𝑎! 𝑏!

𝚥 𝑎! 𝑏!

𝑘 𝑎! 𝑏!

𝑎! 𝑎! 𝑏! − 𝚥 𝑏!

𝑎! 𝑎! 𝑏! + 𝑘 𝑏!

− 𝚥 𝑎! 𝑏! − 𝑎! 𝑏!

𝑎! 𝑏!

+ 𝑘 𝑎! 𝑏! − 𝑎! 𝑏!

𝚤, 𝚥, and 𝑘 are called unit vectors. A unit vector, is a vector of length 1



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!

Magnitude or Length of a vector A bold letter is a vector i.e. 𝑎 = 𝒂 = 𝑎! , 𝑎! , 𝑎! 2𝐷,

𝒂 = 𝑎 =𝑎=

𝑎!! + 𝑎!!

3𝐷,

𝒂 = 𝑎 =𝑎=

𝑎!! + 𝑎!! + 𝑎!!

Unitizing a vector To make the vector be of length 1 but preserve the direction. 𝑎 𝑎! , 𝑎! 2𝐷, 𝑎= = 𝑎 𝑎!! + 𝑎!! 𝑎 𝑎! , 𝑎! , 𝑎! 3𝐷, 𝑎= = 𝑎 𝑎!! + 𝑎!! + 𝑎!! Resultant Vector 𝑅 = 𝑎 + 𝑏 In physics you will be usually be given the vector e.g. (e.g. = for example) 𝑣 (𝑣 = velocity) Copyright © WeSolveThem LLC

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The resultant vector, 𝑣 would be a vector that can be broken into a 𝑥 and 𝑦 component. Angle with respect to x-axis Angle with respect to y-axis



𝑣 = 𝑣 cos 𝜃 , 𝑣 sin 𝜃



𝑣 = 𝑣 sin 𝜙 , 𝑣 cos 𝜙

𝑣,

𝑣! = 𝑣 cos 𝜃 , 𝑣! = 𝑣 cos 𝜃 , 0 𝑣! = 𝑣 sin 𝜃 , 𝑣! = 0, 𝑣 sin 𝜃



𝑣,

𝑣! = 𝑣 sin 𝜙 , 𝑣! = 𝑣 sin 𝜙 , 0 𝑣! = 𝑣 cos 𝜙 , 𝑣! = 0, 𝑣 cos 𝜙

𝑅 = 𝑣 = 𝑣! + 𝑣! = 𝑣 cos 𝜃 , 0 + 0, 𝑣 sin 𝜃 = 𝑣 cos 𝜃 , 𝑣 sin 𝜃

𝑣 =

𝑣 cos 𝜃

!

+ 𝑣 sin 𝜃

!

=

𝑣 ! cos ! 𝜃 + 𝑣 ! sin! 𝜃 =

𝑣 ! cos ! 𝜃 + sin! 𝜃 =

𝑣 ! 1 = 𝑣

This may be slightly confusing with the notation because of the vectors but in physics, you will be ! given a number for the vector i.e. 𝑣 = −25 ! , 𝜃 = 25° (a vector has magnitude and direction, which means it can be 𝑣 =

−25

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! !

cos 25° , −25

! !

!

sin 25° or for magnitude 𝑣 = 𝑣 = 25 ! .

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Quick Reference Arithmetic

𝑎 𝑏 = 𝑎 𝑐 𝑏𝑐

𝑎𝑏 ± 𝑎𝑐 = 𝑎 𝑏 ± 𝑐 = 𝑏 ± 𝑐 𝑎 𝑎−𝑏 𝑏−𝑎 = 𝑐−𝑑 𝑑−𝑐

𝑎𝑏 + 𝑎𝑐 = 𝑏 + 𝑐, 𝑎 ≠ 0 𝑎

𝑎 𝑎 𝑐 𝑎𝑐 = ∙ = 𝑏 1 𝑏 𝑏 𝑐

𝑎! = 𝑎

𝑎! = 1

𝑎! = 𝑎!!! 𝑎! Radicals 𝑎=

𝑎=

!

!"

𝑎=

𝑎 𝑏

!

=

𝑎!! = 𝑎! 𝑏!

!

𝑎 = 𝑎!"

!

𝑎!

=

! 𝑎!

𝑎 𝑏

!

!

!!

1 𝑎! =

𝑏! 𝑎!

𝑎! = 𝑎, 𝑛 𝑖𝑠 𝑜𝑑𝑑

𝑎!

=

! 𝑎!

𝑏 𝑎𝑏 = 𝑐 𝑐

𝑎

𝑎 𝑏 = 𝑎 ∙ 𝑑 = 𝑎𝑑 𝑐 𝑏 𝑐 𝑏𝑐 𝑑

𝑎±𝑏 𝑎 𝑏 = ± 𝑐 𝑐 𝑐

Exponential

! !

𝑎 𝑐 𝑎𝑑 ± 𝑏𝑐 ± = 𝑏 𝑑 𝑏𝑑

1 = 𝑎! 𝑎!! 𝑎!

! !

𝑎! 𝑎! = 𝑎!!! ! !

𝑎!

= 𝑎!

!

!

!

= 𝑎! !

𝑎! = 𝑎 , 𝑛 𝑖𝑠 𝑒𝑣𝑒𝑛 𝑎 = 𝑏

!

𝑎

!

𝑏

!

=

𝑎! ! 𝑏!

𝑎 = 𝑏

! !



Fractions 𝑎 𝑐 𝑎𝑑 ± 𝑏𝑐 𝑔 𝑥 ℎ 𝑥 𝑔 𝑥 𝑟 𝑥 ± 𝑓 𝑥 ℎ 𝑥 ± = ± = 𝑏 𝑑 𝑏𝑑 𝑓 𝑥 𝑟 𝑥 𝑓 𝑥 𝑟 𝑥 Logarithmic ln 𝑏 = log ! 𝑏 𝑦 = log ! 𝑥 ⇔ 𝑥 = 𝑏 ! 𝑒 ≈ 2.72 log ! 𝑎 = 1 ln 𝑎 log ! 1 = 0 log ! 𝑎! = 𝑢 log ! 𝑢 = ln 𝑢 log ! 𝑢! = 𝑏 log ! 𝑢 𝑢 ln 𝑏 log ! 𝑢𝑣 = log ! 𝑢 + log ! 𝑣 log ! = log ! 𝑢 − log ! 𝑣 log ! 𝑏 = 𝑣 ln 𝑎 Copyright © WeSolveThem LLC

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𝑣 = ln 𝑢 ⇒ 𝑢 = 𝑒

! !

ln 𝑎 = undefined, 𝑎 ≤ 0 ln 𝑒 ! = 1 ⇒ 𝑒 !" ! = 1

𝑣=𝑒

!

⇒ 𝑢 = ln 𝑣

𝑒= !!!

ln 𝑒 ! = 𝑢 ⇒ 𝑒 !" ! = 𝑢

ln 1 = 0 ln 𝑢! = 𝑏 ln 𝑢

ln 𝑢𝑣 = ln 𝑢 + ln 𝑣

Quadratic Formula 𝑎𝑥 ! + 𝑏𝑥 + 𝑐 = 0

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1 𝑛!

⇒

𝑥=

ln

𝑢 = ln 𝑢 − ln 𝑣 𝑣

−𝑏 ± 𝑏 ! − 4𝑎𝑐 2𝑎

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