1. Mathematics

2. Differential Equations - 1
Session
Differential Equations - 1
I am going to take a session in mathematics , the topic is
Linear equation in two variables.

3. 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

4. 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.

5. Examples

6. 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

7. 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.

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

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

10. 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

11. 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.

12. 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.

13. 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

14. is the general solution of the differential equation
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.

15. is a particular solution of the differential equation
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

16. Example - 4

17. Solution Cont.

18. 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.

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

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

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

22. 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]

23. 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.

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

25. 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.
Solution: The general equation of a circle is
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).

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

27. Solution Cont.

28. Thank you