Mathematics

Session Differential Equations - 1

Session Objectives  Differential Equation  Order and Degree  Solution of a Differential Equation, General and Particular Solution  Initial Value Problems  Formation of Differential Equations  Class Exercise

Differential Equation An equation containing an independent variable x,dependent variable y and the differential coefficients of the dependent variable y with respect to independent variable x, i.e.

Examples

Order of the Differential Equation The order of a differential equation is the order of the highest order derivative occurring in the differential equation. The order of the highest order derivative Therefore, order is 2

Degree of the Differential Equation The degree of a differential equation is the degree of the highest order derivative, when differential coefficients are made free from fractions and radicals. The degree of the highest order derivative is 2. Therefore, degree is 2.

Example - 1 Determine the order and degree of the differential equation:

Solution Cont. The order of the highest derivative is 1 and its degree is 2.

Example - 2 Determine the order and degree of the differential equation: Solution: We have Here, the order of the highest order is 4 and, the degree of the highest order is 2

Linear and Non-Linear Differential Equation A differential equation in which the dependent variable y and its differential coefficients i.e. occur only in the first degree and are not multiplied together is called a linear differential equation. Otherwise, it is a non-linear differential equation.

Example - 3 is a linear differential equation of order 2 and degree 1. is a non-linear differential equation because the dependent variable y and its derivative are multiplied together.

Solution of a Differential Equation The solution of a differential equation is the relation between the variables, not taking the differential coefficients, satisfying the given differential equation and containing as many arbitrary constants as its order is. For example: is a solution of the differential equation

General Solution If the solution of a differential equation of nth order contains n arbitrary constants, the solution is called the general solution. is the general solution of the differential equation is not the general solution as it contains one arbitrary constant.

Particular Solution A solution obtained by giving particular values to the arbitrary constants in general solution is called particular solution. is a particular solution of the differential equation

Example - 4

Solution Cont.

Initial Value Problems The problem in which we find the solution of the differential equation that satisfies some prescribed initial conditions, is called initial value problem.

Example - 5 satisfies the differential equation Show that is the solution of the initial value problem

Solution Cont. is the solution of the initial value problem.

Formation of Differential Equations Assume the family of straight lines represented by is a differential equation of the first order. X Y O

Formation of Differential Equations Assume the family of curves represented by where A and B are arbitrary constants. [Differentiating (i) w.r.t. x] [Differentiating (ii) w.r.t. x]

Formation of Differential Equations [Using (i)] is a differential equation of second order Similarly, by eliminating three arbitrary constants, a differential equation of third order is obtained. Hence, by eliminating n arbitrary constants, a differential equation of nth order is obtained.

Example - 6 Form the differential equation of the family of curves a and c being parameters. Solution: We have is the required differential equation. [Differentiating w.r.t. x]

Example - 7 Find the differential equation of the family of all the circles, which passes through the origin and whose centre lies on the y-axis. If it passes through (0, 0), we get c = 0 This is an equation of a circle with centre (- g, - f) and passing through (0, 0). Solution: The general equation of a circle is

Solution Cont. Now if centre lies on y-axis, then g = 0. This represents the required family of circles.

Solution Cont.

Thank you