### Abstract:

Differential equation is mathematical equation on for an unknown function of one or more serval variable that relateds,the value of function itself and its derivatives of various order. Differential equation plays a prominent role in engineering, physics economics and other disciplines. An ordinary differential equation (ODE) is differential equation which the unknown function (also known as depend variable)is function of single independent variable in the simplest form the unknown function is real or complex valued function but more generally ,it may be vector valued or matrix valued. In general it is not possible to solve the second order and higher order linear ordinary differential equation. That is we will examine equations that have special forms ,whichallow us to either reduce the order of equation or transform it to some known equation which is easy to solve . This gives us superficially different methods for solving. Some time it is possible to solve anon linear equation by making a change of dependent variable that converts it into alinear equation .One of the most important such equations which use for transform are(Euler ,Riccati,Bernoulli) Riccati Equation it can factoring second order opertorss consider the order linear then we would be able to solve the differential equation factoring reduces the problem to a system of first order equation of Differential equation that are invariant under the change of variable x=c(ζ)familiar example form linear equation ,we note that Euler equation equidimensional -in-x Written then new derivations under the change of variable Anequation is scale invarinant if it is invarinant under the change of variable x=c(ζ), y(x)=cv(ζ)for some value an equidimenstional in-x equation with the change variables y(x)=xv(ζ) . The study recommend use these Techniques for solving some Complicated nonlinear ordinary differential equations.

### Description:

A Dissertation Submitted to the University of Gezira in Partial Fulfillment of the Requirements for the Aword of the Degree of Master of Science in Mathematics, Department of Mathematics, Faculty of Mathematical and Computer Sciences, September 2015