Differential Equations

SYNOPSIS

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

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
    .

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
     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:

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 n^th 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  , which is in the variable separable form, is given by
    .

16. A differential equation of the form   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  = f(x, y) or  = 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:

    … (1)
    Substitute y = v.x … (2)
    Differentiate (2) with respect to x
    … (3)
    Substitute and separate the variables
    
    Integrate 

21. A differential equation of the form  , 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  , where P₁ and Q₁ are constants or functions of y only, is called a first-order linear differential equation.

22. Solution of  , 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  , where P₁ and Q₁ are constants or functions of y:
    Integrating factor (IF) = e ∫^P₁^dy
    Solution: x (IF) = ∫ (Q₁ × IF)dy + C