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JEE Maths Differential Equations

Differential Equations PDF Notes, Important Questions and Synopsis

SYNOPSIS

  1. An equation involving derivatives of a dependent variable with respect to an independent variable is called a differential equation.
    Example: begin mathsize 12px style dy over dx equals cosx end stylebegin mathsize 12px style 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 number of the highest order derivative occurring in the differential equation.
    Example: Order of the differential equation
     begin mathsize 12px style fraction numerator straight d cubed straight y over denominator dx cubed end fraction plus 3 straight x open parentheses dy over dx close parentheses minus 8 straight y equals 0 end style.

  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 straight d squared straight y over denominator dx squared end fraction close parentheses cubed plus left parenthesis straight c plus straight b right parenthesis open parentheses dy over dx close parentheses to the power of 4 equals straight y end style is 3.

  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 expressed 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 straight n minus 1 end exponent straight y over denominator dx to the power of straight n minus 1 end exponent end fraction plus straight P subscript 2 fraction numerator straight d to the power of straight n minus 2 end exponent straight y over denominator dx to the power of straight n minus 2 end exponent end fraction 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 end cell row cell where text  P end text subscript text 0 end text end subscript comma straight P subscript 1 comma straight P subscript 2 comma. .. comma straight P subscript straight n minus 1 end subscript comma straight P subscript straight n text  and Q are either  end text end cell row cell text constants or functions of independent variable x. end text end cell end table end style

  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 an independent variable, dependent variable and constants.
    2. Determine the number of constants and assume there be 'n' arbitrary constants.
    3. Differentiate the relation n times with respect to x, where n is number of arbitrary constants.
  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: 

    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 a linear differential equation.
  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 open parentheses straight x close parentheses dx equals straight g open parentheses straight y close parentheses dy end style , which is in the variable separable form, is given by
    begin mathsize 12px style integral straight f open parentheses straight x close parentheses dx equals integral straight g open parentheses straight y close parentheses dy plus straight C comma space where space straight C space is space any space arbitrary space constant end style.

  16. A differential equation of the form  begin mathsize 12px style dy over dx equals straight f open parentheses ax plus by plus straight c close parentheses end style can be reduced to the variable separable form by substituting ax + by + c = v.

  17. A differential equation which can be expressed in the form begin mathsize 12px style dy over dx end style = f(x, y) or  begin mathsize 12px style dx over dy end style= g(x, y), where f(x, y) and g(x, y) are homogeneous functions, is called a homogeneous 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:

    begin mathsize 12px style dy over dx equals straight F left parenthesis straight x comma straight y right parenthesis equals straight g open parentheses straight y over straight x close parentheses end style… (1)
    Substitute y = v.x … (2)
    Differentiate (2) with respect to x
    begin mathsize 12px style dy over dx equals straight v plus straight x dv over dx end style… (3)
    Substitute and separate the variables
    begin mathsize 12px style fraction numerator dv over denominator straight g left parenthesis straight v right parenthesis minus straight v end fraction equals dx over straight x end style
    Integrate begin mathsize 12px style integral fraction numerator dv over denominator straight g left parenthesis straight v right parenthesis minus straight v end fraction equals integral dx over straight x text   end text plus straight C end style

  21. A differential equation of the form begin mathsize 12px style dy over dx plus Py equals straight Q end style , where P and Q are constants or functions of x only, is called a first-order linear differential equation.

    OR

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

  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 (IF) = e ∫Pdx
    Solution: y (IF) = ∫ (Q × IF)dx + C

  23. Solution of begin mathsize 12px style dx over dy plus straight P subscript 1 straight y equals straight Q subscript 1 end style , where P1 and Q1 are constants or functions of y:
    Integrating factor (IF) = e ∫P1dy
    Solution: x (IF) = ∫ (Q1 × IF)dy + C