An alternative pedagogical design is discussed that aims to guide engineering students solve first-order ordinary differential equations (ODEs), and based on students’ learning weaknesses identified from previous teaching activities. This approach supported student’s self-enrichment through exploration of relevant resources in ODEs, guided towards the choice their own effective ways for solving...

A semi-algorithm to find elementary first order invariants of rational second order ordinary differential equations a,b Abstract Here we present a method to find elementary first integrals of rational second order ordinary differential equations (SOODEs) based on a Darboux type procedure [16, 12, 13]. Apart from practical computational considerations, the method will be capable of telling us (u...

This part is concerned with the numerical solution of initial value problems for systems of ordinary differential equations. We will introduce the most basic one-step methods, beginning with the most basic Euler scheme, and working up to the extremely popular Runge–Kutta fourth order method that can be successfully employed in most situations. We end with a brief discussion of stiff differentia...

A semi-algorithm to find elementary first order invariants of rational second order ordinary differential equations a,b Abstract Here we present a method to find elementary first integrals of rational second order ordinary differential equations (SOODEs) based on a Darboux type procedure [16, 12, 13]. Apart from practical computational considerations, the method will be capable of telling us (u...

2. General Remarks Second order ODEs are much harder to solve than first order ODEs. First of all, a second order linear ODE has two linearly independent solutions and a general solution is a linear combination of these two solutions. In addition, many popular second order ODEs have singular points. Except for a few cases, the solutions have no simple mathematical expression. However, there is ...

in this paper, the chebyshev spectral collocation method(cscm) for one-dimensional linear hyperbolic telegraph equation is presented. chebyshev spectral collocation method have become very useful in providing highly accurate solutions to partial differential equations. a straightforward implementation of these methods involves the use of spectral differentiation matrices. firstly, we transform ...