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# Differential Equations

## 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.

2. Partial Differential Equation

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

e.g.  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)

is order m and degree p.

Note that

In the differential equation  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.

Example

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

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

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 :

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  which is the variable separated form, is given by,
Where C is any orbitrary constant.

16.  A differential equation of the from  can be reduced to the variable separated from by substituting  .

17. A differential equation which can be expressed in the from  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

21. A differential equation of the from, 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  , then P1 and Q1 are constants or functions of y.

22. Solution of  where P and Q are constants or functions of X:
Integrating factor
Solution:

23. Solution of  where P and Q are constants or functions of X:
Integra ting factor
Solution