Keywords of Statics

Types of Statistics

1. Inductive statistics, or statical inference :- condition under inference

2. descriptive or deductive statistics.

Variabls:

1. discrete

2. continue

Frequency

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frequance distribution, class interval, class limits, 

class mark : mid point of class

relative frequency distribution : precentage distribution or relative frequency table.  : frequency of class divided by total frequency

cumulative frequency and ogives: total frequency upto class upper boundary.  "or more", "or less"

relative cumulative frequency : cumulative frequency divided by total frequency.

Averages or Measures of Centeral Tendency

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1. Arithmetic mean
2. Median : midle value
3. Mode  : greatest frequency
4. Geometric Mean  : Nth root of multiplication
5. Harmonic mean  : reciprocal arthmatic mean
6. Weighted Arithmetic mean ; arthmatic mean multiply with weight
7. Root Mean Square(RMS) or quadratic mean
8. Quartiles, Deciles and Percentiles

           Mean-mode = 3(mean-median)

The Standard Deviations

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1. Dispersion, Variation of the data
2. Range
3. Mean Deviation
4. Standard Deviation : Root mean square deviation
5. Variance : square of standard deviation

1. Moments
2. Skewness
3. Kurtosis

Second Order Differential Equation

Typical form of second order differential equation

                 y"=f(x,y,y')

Solutions

1. y' and y missing :   y"=f(x) integrate twice with respect to x
2. y missing : y"=f(y',x) 
1. put p=y'    =>  p'=y"=f(p,x) this become first order equation
3. y' and x missing : y"=f(y)     => put p=y'   =>    => this is seprable with respect to p and y.
4. x missing   : y"=f(y',y) => put p=y' =>   

Second order linear homogeneous  equation with constant coefficients 

1. form(characteristic) : y"+a₁y'+a₀y=0   => λ²+a₁λ+a₀=0  
1. Solving this quadratic equation got 3 kind of roots.
1. real and distinct λ1,λ2 then solution
1. y=c₁e^(λ1x)+c₂e^(λ2x)
2. λ₁=a+ib and λ₂=a-ib complex conjugates
1. y=c₁e^ax cos(bx)+c₂e^ax sin(bx)
3. λ1=λ2
1. y=c₁e^λ1x+c₂x^λ1x
2. equation should be linear,  homogeneous  and constant coefficient  only.
3. This solution also workout for similar n^th order equations.

Non-Homogeneous Equations

        L(y)=ϕ(x)  then General solution GS  y=y_h+y_p  =>   y=y_c+y_p

there is three way to find y_p

1. "Guesswork"
2. Annihilator
3. D-operator

Guesswork Method

1. Polynomial function y_p=g(x)=p₀+p₁x+p₂x²+........+p_nxⁿ  Solve this equation with differentiating wrt x and equates coefficients of xⁿ
2. exponential function  y_p=g(x)=Ae^ax      Solve this equation with differentiating wrt x
3.  Trigonometrical function    y_p=g(x)=A sin βx + B cos βx    Solve this equation with differentiating wrt x
4. Above 3 will work for nth order differential equation.

Eluer equation : linear ODE with variable coefficients

            use y=x^λ

First Order Differential Equation

Differential Equation

1. Ordinary differential Equation(ODE)
2. Partial Differential Equation(PDE)

Property

1. Order of DE
2. Degree of DE

Condition for DE

1. Initial value 
2. Boundary value

Type of Differential Equation

1. Linear and non-linear
2. Homogeneous and non-homogeneous
3. First order differential equation
4. Second order differential equation
5. n-th order differential equation.
6. Bernoulli equation

First order differential equation

1. Standard form of FODE   y'=f(x,y)
2. Differential form M(x,y)dx+N(x,y)dy=0
3. linear form y'+p(x)y=q(x)
4. Bernoulli y'+p(x)y=q(x)yⁿn
5. Homogeneous f(x,y)=f(tx,ty)
6. exact differential equation M_y(x,y)=N_x(x,y)
1. 

Solution 4 first ODE

1. Separable variables f(x)dx+g(y)dy=0
2. Reduction of Homogeneous 
1. substitute y=vx => y' = v+xv'   OR x=uy => x'=u+xu'
3. Exact differential equation
1. do integration g_x(x)=M(x,y) with respect to x => g(x,y)= I(M(x,y)dx+h(y)
2.  When do integration constant would depends on y function h(y)
3. this result need to do partial differential with respect to y and put equal to g_y(x,y)=N(x,y)
4. Integrate this and get solution for h(y) and put in step 1 equation.
4. Linear Differential equation. y'+p(x)y=q(x)
1. find integration factor
1. IF(x,y)=ePower(I(px)dx)
2. mulitply this integration factore with equation get this
1. yIF=I(IF*q(x))dx + C
5. Reduction of Bernoulli Equations
1. substitute z=y^1-n
2. Now we have linear equation.