Dictionary Rank of a Word | Permutations & Combinations

 PERMUTATIONS & COMBINATIONS Rank of the word or Dictionary order of the English words like COMPUTER, COLLEGE, SUCCESS, SOCCER, RAIN, FATHER, etc. Dictionary Rank of a Word Method of finding the Rank (Dictionary Order) of the word  “R A I N” Given word: R A I N Total letters = 4 Letters in alphabetical order: A, I, N, R No. of words formed starting with A = 3! = 6 No. of words formed starting with I = 3! = 6 No. of words formed starting with N = 3! = 6 After N there is R which is required R ----- Required A ---- Required I ---- Required N ---- Required RAIN ----- 1 word   RANK OF THE WORD “R A I N” A….. = 3! = 6 I……. = 3! = 6 N….. = 3! = 6 R…A…I…N = 1 word 6 6 6 1 TOTAL 19 Rank of “R A I N” is 19 Method of finding the Rank (Dictionary Order) of the word  “F A T H E R” Given word is :  "F A T H E R" In alphabetical order: A, E, F, H, R, T Words beginni

Differential Equations Chapter 9 Class XII

Differential Equations Chapter 9 Class 12
Basic concepts and formulas of differential equations chapter 9 class 12, method of solving homogeneous differential equations and linear differential equations.

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Differential equation

An equation involving derivative of the dependent variable with respect to independent variable is called a differential equation.
Differential equations are widely used in the field of Physics, Chemistry, Biology, Geology, Economics etc.
 Examples of differential equations:\[\frac{dy}{dx}+y=5\]\[\frac{d^{4}y}{dx^{4}}+5\frac{d^{2}y}{dx^{2}}+ 3y=10\]
In this chapter we will discuss the following points
  • Basic concepts related to the differential equations.
  • General and particular solutions of differential equations.
  • Formation of differential equations.
  • Method of solving Homogeneous Differential Equations
  • Method of solving Linear Differential Equations .
  • Some applications of differential equations in different areas.
Special notations used for derivatives are as follows\[\frac{dy}{dx}=y'=y_{1}\]\[\frac{d^{2}y}{dx^{2}}=y''=y_{2}\]\[\frac{d^{3}y}{dx^{3}}=y'''=y_{3}\]\[\frac{d^{4}y}{dx^{4}}=y''''=y_{4},\: \: \: ..... \: and\: so\: on\]

Order of a differential equation

Order of a differential equation is defined as the order of the highest order derivative of the dependent variable with respect to the independent variable involved in the given differential equation.\[Eqn.1)..\frac{dy}{dx}+y=0\Rightarrow order\: of\: the\: D.E.\: is\: \: one\]\[Eqn.2)..\left (\frac{d^{2}y}{dx^{2}} \right )^{5}+y=0 \Rightarrow order\: of\: the\: D.E.\: is\: \: two\]\[Eqn.3)..\left (\frac{d^{3}y}{dx^{3}} \right )^{2}+\frac{d^{4}y}{dx^{4}}+y=0\Rightarrow order\: of\: the\: D.E.\: is\: \: four\]

Degree of differential equation

It is defined as the degree(or power) of the highest order derivative involved in the given differential equation.
Degree of a differential equation is defined only if the differential equation is the polynomial equation in y'.
Order of every D.E. exist but degree may or may not be exist.
Order and degree if exist are always positive.
Degree of differential eqn.1) is  one.
Degree of differential eqn.2) is  five.
Degree of differential eqn.3) is  one.

Degree of the following differential equations is not exist.\[\frac{dy}{dx}+sin\left (\frac{dy}{dx} \right )+1\]\[\frac{d^{3}y}{dx^{3}}+log\left (\frac{dy}{dx} \right )+1\] \[y'''+y^{2}+e^{y'}\]In all these differential equations we can find the order but degree is not exist. Because these equations are not the polynomial equations in its derivatives.\[\left (\frac{d^{3}y}{dx^{3}} \right )^{4}+\left ( \frac{dy}{dx} \right )^{2}+e^{\frac{dy}{dx}}+5=0\]In the above equation order is 3 but degree is not exist.


General Solution and Particular Solution of the differential equation.

Value of dependent variable(y) which satisfy the given differential equation is called solution of that D. E.
Solutions are of two types
1) General Solution:
Solution with the arbitrary constants a, b, c.....etc. is called the general solution of the differential equation.
2) Particular Solution:
Solution of the D.E. obtained after eliminating the arbitrary constants is called particular solution.

Note: Number of arbitrary constants is equal to the order of the differential equation.
For example:
If order of the differential equation is two then number of arbitrary constants is also two.
If order of the differential equation is five then number of arbitrary constants is also five.

Different Methods of Solving Differential Equations

There are mainly three methods of solving Differential Equations
1) Variable Separable Method
2) Homogeneous Differential Equations.
3) Linear Differential Equations.
Explanations of Methods of solving Differential Equations

1) Variable Separable Method

i) Write the differential equation.
ii) Bring all the terms with variable y to the LHS and bring all the terms with variable x to the RHS.
iii) Bring the negative sign if any to the RHS.
iv) Integrating on both sides and get General Solution.
v) For particular solution putting the given values of x and y in the General Solution and find the value of C (Constant).
vi) Putting the value of C in the general solution to get the particular solution.  

2) Homogeneous Differential Equations

If dy/dx=F(x, y) is a differential equation, then it is called a homogeneous differential equation if it can be put in the form f(y/x) of degree zero.
In other words\[If\:\; \frac{dy}{dx}=F(x,y)=f\left (\frac{y}{x} \right ) \; of\; order\; \; zero\]Then F(x, y) is called the homogeneous differential equation.

Method of solving H.D.E.
i) Put y/x = v ⇒ y = vx
ii) Differentiating w.r.t. x on both side  we get \[\frac{dy}{dx}=v+x\frac{dv}{dx}\]
iii) Putting the values from eqn.(i) and eqn.(ii) in the given homogeneous differential equation.
iv) Now separate the variable v and x.
v) Integrating on both side w. r. t. x we get the value of v in terms of x and arbitrary constants.
vi) Putting v = y/x and then find the value of y. This is called the general solution of H.D.E.
vii). To find the particular solution from General solution we eliminate the arbitrary constants.

Other forms of Homogeneous Differential Equation
Other form of HDE is given as\[\frac{dx}{dy}=F(x,y)= f\left ( \frac{x}{y} \right )\]In such type of equations we put x/y = v ⇒ x = vy and then differentiating w.r.t y on both side. Then follow the sequence of steps as given above to find the value of x.

3) Linear Differential Equation(LDE)

An equation of the following type is called Linear Differential Equation.\[\frac{dy}{dx}+Py=Q\]Where P and Q are either constants or the functions of x only and the coefficient of dy/dx should be unity.

Method of solving LDE
i) Make the coefficient of dy/dx unity (or 1) and then check whether it is a LDE or not.
ii) Find the Integrating Factor (IF)\[IF=e^{\int Pdx}\]
iii) Multiplying on both side by IF\[IF\times \frac{dy}{dx}+IF\times y=IF\times Q\]\[\frac{d}{dx}\left ( y\times IF \right )=Q\times IF\]
iv) Integrating on both side we get\[\int \frac{d}{dx}\left ( y\times IF \right )dx=\int \left (Q\times IF \right )dx+C\]\[y\times IF=\int \left (Q\times IF \right )dx+C\]From this equation we can find the value of y.

Other form of Linear differential equation\[\frac{dx}{dy}+Px=Q\] Where P and Q are either constants or the functions of y only. In this case Integrating Factor(IF) can be calculated as \[I.F.=e^{\int Pdy}\]The solution of LDE is given as \[x\times IF=\int \left (Q\times IF \right )dy+C\]

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