1. Differential Equations
Represented by
Dr. Shorouk Ossama

2. Ordinary Differential Equations
If we replace on the right-hand side of the last equation by the symbol y, the derivative becomes:

3. DEFINITION:
An equation containing the derivatives of one or more dependent variables, with respect to one or more independent variables, is said to be a differential Equation (DE).

4. classify differential equations by
Classification By Type:
If an equation contains only ordinary derivatives of one or more dependent variables with respect to a single independent variable it is said to be an Ordinary Differential Equation (ODE). For example,

5. An equation involving partial derivatives of one or more dependent variables of two or more independent variables is called a partial differential equation (PDE). For example:

6. Ordinary derivatives will be written by using either the Leibniz notation dy/dx, d2y/dx2, d3y/dx3,... or the prime notation y`, y``, y```,…. In general, the nth derivative of y is written dny/dxn or y(n).

7. Classification By Order :
The order of a differential equation (ODE) is the order of the highest derivative in the equation. For example,
is a second-order ordinary differential equation.

8. The highest derivative y(n) in terms of the remaining n + 1 variables
The highest derivative y(n) in terms of the remaining n + 1 variables. The differential equation:

10. Classification By Linearity:
An nth-order ordinary differential equation is said to be linear if F is linear in y, y`,...,y(n). This means that an nth-order ODE is linear.
. For example,

11. Linear
Non Linear

12. Initial Value Problem
We are often interested in problems in which we seek a solution y(x) of a differential equation so that y(x) satisfies prescribed side conditions—that is, conditions imposed on the unknown y(x) or its derivatives. On some interval I containing x0 the problem

13. are called initial conditions.
Where: y0, y1,...,y n-1 are specified real constants, is called an initial-value problem (IVP). The values of y(x) and its first n-1 derivatives at a single point x0,
y(x0) = y0,
y`(x0) = y1,...,
y(n-1) (x0) = yn-1 ,
are called initial conditions.

15. A solution y(x) of the differential equation y`= f (x, y) on an interval I containing x0, so that its graph passes through the specified point (x0, y0).

16. A solution y(x) of the differential equation y``
A solution y(x) of the differential equation y`` = f (x, y, y`) on an interval I containing x0, so that its graph not only passes through (x0,y0) but the slope of the curve at this point is the number y0.

17. Example: consider the ordinary differential equation y`` + y = 0 to be solved for the unknown y(x). Subject to the conditions y(0) = 0, y`(0) = 5 Solution: Without the initial condition, the general solution to this equation is: Y(x) = A sinx + B cosx

18. From the initial condition y(0) = 0 one obtains 0 = A. 0 + B
From the initial condition y(0) = 0 one obtains 0 = A.0 + B.1, which implies that B=0. From the initial condition then the solution will be y(x) = A sinx
from the initial condition y`(0) = 5 , we have y`(x) = A cosx and y`(0) = A cos0 = 5 , so y(x) = 5 sinx

19. Solution of the Differential Equation
Consider the differential equation
A function y = f(x) is called a solution of the differential equation if y = f(x) satisfies the differential equation.

20. Example: Show that function y = 2x + 6x2 is a solution of y` (x + 3x2) – y (1 + 6x) = 0 Solution Since y` = x, substitute in the differential equation then (2 + 12x) (x + 3x2) – (2x + 6x2 ) (1 + 6x) = 0 So the L.H.S = R.H.S = 0

21. SUMMARY
From Pages 6 To 11
Page 12: Exercises

22. Thanks