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Differential Equations

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Differential Equations PDF Notes, Important Questions and Formulas

Differential Equation

DEFINITION:

An equation that involves independent and dependent variables and the derivatives of the dependent variables is called a DIFFERENTIAL EQUATION.

There are two types of differential equation:

1. Ordinary Differential Equation:

A differential equation is said to be ordinary, if the differential coefficients have reference to a single independent variable only

e.g. begin mathsize 12px style fraction numerator straight d squared straight y over denominator dx squared end fraction minus fraction numerator 2 dy over denominator dx end fraction plus cosx equals 0 end style

2. Partial Differential Equation

A differential equation is said to be partial, if there are two or more independent variables,

e.g. begin mathsize 12px style fraction numerator partial differential straight u over denominator partial differential straight x end fraction plus fraction numerator partial differential straight u over denominator partial differential straight y end fraction plus fraction numerator partial differential straight u over denominator partial differential straight z end fraction equals 0 end style is a partial differential equation. We are concerned with ordinary

differential equation only.

 

SOLUTION (PRIMITIVE) OF DIFF.EQUATION

Finding the unknown function which satisfies given differential equation is called SOLVING OR INTEGRATING the differential equation. The solution of the differential equation is also called its PRIMITIVE, because the differential equation can be regarded as a relation derived from it.

 

ORDER OF DIFFERENTIAL EQUATION

The order of a differential equation is the order of the highest differential coefficient occurring in it.

 

DEGREE OF DIFFERENTIAL EQUATION

The degree of a differential equation which can be written as a polynomial in the derivatives is the degree of the derivative of the highest order occurring in it, after it has been expressed in a form which is free from radicals and fractions so far as derivatives are concerned, thus the differential equation (x,y)

 

begin mathsize 12px style open square brackets fraction numerator begin display style straight d to the power of straight m straight y end style over denominator begin display style dx to the power of straight m end style end fraction close square brackets to the power of straight p plus straight ϕ left parenthesis straight x comma straight y right parenthesis open square brackets fraction numerator begin display style straight d to the power of straight m minus 1 end exponent left parenthesis straight y right parenthesis end style over denominator begin display style dx to the power of straight m minus 1 end exponent end style end fraction close square brackets to the power of straight q plus. ... equals straight o end style is order m and degree p.

Note that

In the differential equation begin mathsize 12px style straight e subscript straight y " minus xy " plus straight y equals 0 end style order is three but degree doesn’t apply.

FORMATION OF DIFFRENTIAL EQUATION

If an equation with independent and dependent variables having some arbitrary constant is given, then a differential equation is obtained as follows:

  1. Differentiate the given equation w.r.t. the independent variable (say x) as many times as the number of arbitrary constants in it.
  2. Eliminate the arbitrary constants. The eliminant is the required differential equation.

 

Remark

A differential equation represents a family of curves all satisfying some common properties. This can be considered as the geometrical interpretation of the differential equation.

 

Differential Equations

 

  1. An equation involving derivatives of a dependent variable with respect to an independent variable is called a differential equation.

    Examplebegin mathsize 12px style dy over dx equals cosx dy over dx equals fraction numerator straight x squared plus straight y squared over denominator 2 straight x end fraction end style

  2. The order of a differential equation is the order of the highest order derivative occurring in the differential equation.     Example: Order of 

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  3. The degree of a differential equation is the highest power (exponent) of the highest order derivative in it, when it is written as a polynomial in differential coefficients.

    Degree of equation begin mathsize 12px style open parentheses fraction numerator begin display style straight d squared straight y end style over denominator begin display style dx squared end style end fraction close parentheses cubed plus left parenthesis straight c plus straight b right parenthesis open parentheses fraction numerator begin display style dy end style over denominator begin display style dx end style end fraction close parentheses to the power of 4 equals straight y text  is  end text 3 end style

  4.  The order and the degree of a differential equation are positive integers.

  5. A differential equation is a linear differential equation if it is expressible in the form :

    begin mathsize 12px style table attributes columnalign left end attributes row cell straight P subscript 0 fraction numerator straight d to the power of straight n straight y over denominator dx to the power of straight n end fraction plus straight P subscript 1 fraction numerator straight d to the power of blank to the power of straight n minus 1 end exponent end exponent straight y over denominator dx to the power of straight n minus 1 end exponent end fraction plus straight P subscript 2 straight d to the power of straight n minus 2 end exponent over dx to the power of straight n minus 2 end exponent plus. .. plus straight P subscript straight n minus 1 end subscript dy over dx plus straight p subscript straight n straight y equals straight Q comma text   end text end cell row cell text Where P end text subscript 0 comma straight P subscript 1 comma straight P subscript 2 comma. .. comma straight P subscript straight n minus 1 comma end subscript straight P subscript straight n text  and Q are either end text end cell end table end style
    Constants or functions of independent variable x.

  6. A differential equation will be a non-linear differential equation if

    1. Its degree is more than one.
    2. The exponent of any differential equation is more than one.
    3. The exponent of the dependent variable is more than one.
    4. The products containing the dependent variable and its differential coefficients are present.
  7. A function which satisfies a given differential equation is called its solution.

  8. To formulate a differential equation:

    1. Consider an equation involving independent variable, dependent variable and constants.
    2. Determine the number of constants and assume there be 'n' arbitrary constants.
    3. Differentiate the relation in (i) n times with respect to x.
  9. The solution which contains as many arbitrary constants as the order of the differential equation is called a general solution.

  10. The solution which is free from arbitrary constants is called a particular solution.

  11. The order of a differential equation is equal to the number of arbitrary constants present in the general solution.

  12. An nth order differential equation represents an n-parameter family of curves.

  13. Three methods of solving first order and first degree differential equations are

    1. Separating the variables if the variables can be separable.
    2. Substitution if the equation is homogeneous.
    3. Using an integrating factor if the equation is linear different.
  14. The variable separable method is used to solve equations in which variables can be separated, i.e. the terms containing y should remain with dy and the terms containing x should remain with dx.

  15. The solution of the differential equation begin mathsize 12px style straight f left parenthesis straight x right parenthesis dx equals straight g left parenthesis straight y right parenthesis dy comma end style which is the variable separated form, is given by,
    begin mathsize 12px style integral straight f left parenthesis straight x right parenthesis dx equals integral straight g left parenthesis straight y right parenthesis dy plus straight C comma end styleWhere C is any orbitrary constant.

  16.  A differential equation of the from begin mathsize 12px style dy over dx equals straight f left parenthesis ax plus by plus straight c right parenthesis end style can be reduced to the variable separated from by substituting begin mathsize 12px style ax plus by plus straight c equals straight v end style .

  17. A differential equation which can be expressed in the from begin mathsize 12px style dy over dx equals straight f left parenthesis straight x comma straight y right parenthesis or dx over dy equals straight g left parenthesis straight x comma straight y right parenthesis comma where straight space straight f left parenthesis straight x comma straight y right parenthesis space and space straight g left parenthesis straight x comma straight y right parenthesis end style are homogenous differential equation.

  18. The degree of each term is the same in a homogeneous differential equation.

  19. Homogeneous equations can be reduced to the variable separable form by the  substitution of y = vx or x = vy.

  20. Steps to solve a homogeneous differential equation
    

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  21. A differential equation of the frombegin mathsize 12px style dy over dx plus py equals straight Q end style, where P and Q are constant or functions of X, is called a first order linear differential equation. If the equation is of the form begin mathsize 12px style dx over dy plus straight p subscript 1 straight x equals straight Q subscript 1 end style , then P1 and Q1 are constants or functions of y.

  22. Solution of begin mathsize 12px style dy over dx plus Py equals straight Q end style where P and Q are constants or functions of X:
    Integrating factor begin mathsize 12px style left parenthesis IF right parenthesis equals straight e to the power of integral pdx end exponent end style
    Solution: begin mathsize 12px style straight y left parenthesis IF right parenthesis equals integral left parenthesis straight Q plus IF right parenthesis dx plus straight C end style

  23. Solution of begin mathsize 12px style dy over dx plus Py equals straight Q end style where P and Q are constants or functions of X:
    Integra ting factor begin mathsize 12px style left parenthesis IF right parenthesis equals straight e to the power of integral straight p subscript 1 dy end exponent end style
    Solution begin mathsize 12px style straight x left parenthesis IF right parenthesis equals integral left parenthesis straight Q subscript 1 plus IF right parenthesis dy plus straight C end style

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