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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:
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
Example:
The order of a differential equation is the order of the highest order derivative occurring in the differential equation. Example: Order of
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
The order and the degree of a differential equation are positive integers.
A differential equation is a linear differential equation if it is expressible in the form :
Constants or functions of independent variable x.
A differential equation will be a non-linear differential equation if
A function which satisfies a given differential equation is called its solution.
To formulate a differential equation:
The solution which contains as many arbitrary constants as the order of the differential equation is called a general solution.
The solution which is free from arbitrary constants is called a particular solution.
The order of a differential equation is equal to the number of arbitrary constants present in the general solution.
An nth order differential equation represents an n-parameter family of curves.
Three methods of solving first order and first degree differential equations are
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.
The solution of the differential equation which is the variable separated form, is given by,
Where C is any orbitrary constant.
A differential equation of the from can be reduced to the variable separated from by substituting .
A differential equation which can be expressed in the from are homogenous differential equation.
The degree of each term is the same in a homogeneous differential equation.
Homogeneous equations can be reduced to the variable separable form by the substitution of y = vx or x = vy.
Steps to solve a homogeneous differential equation
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 P_{1} and Q_{1} are constants or functions of y.
Solution of where P and Q are constants or functions of X:
Integrating factor
Solution:
Solution of where P and Q are constants or functions of X:
Integra ting factor
Solution
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