SOLUSI DARI PERSAMAAN CAUCHY–EULER NONHOMOGEN KASUS LOGARITMIK

Ordinary differential equation is one form of differential equations that are often found in everyday life. One form of ordinary differential equations which has non–constant coefficients is the Cauchy–Euler differential equation. In the nonhomogeneous Cauchy–Euler differential equations, the undetermined coefficient and the parameter variation were the most method that often used to find the particular solution. This paper aimed to show a new solution that was shorter than the previous methods for nonhomogeneous Cauchy–Euler differential equations with the right side was a logarithmic form. The new solution had been proven to produce the same solution as the ordinary solution sought using the undetermined coefficient method.

NONPERTURBATIVE SOLUTIONS OF THE SD EQUATION FOR AN ABELIAN GAUGE FIELD THEORY

In this letter we present a mechanism in which new nonperturbative solutions of quantum electrodynamics in four dimensions is found. Two nonperturbative solutions are found for approximate Schwinger–Dyson equations. The mass ratio of the three solutions (one of them is the ordinary solution) is approximately 1: (8π/α)2/3 : 8π/α, which is close to the realistic mass ratio of e-, μ- and τ-.

The internal flow problem in axi-symmetric supersonic flow

A scheme of approximate solution is presented for the supersonic flow in a circular duct of slowly varying cross-section for the cases when the conventional linearized theory fails. This happens whenever there are regions in the flow field, termed wave fronts, where the velocity gradients are large in comparison with the variations in velocity. A careful discussion suggests that a valid first approximation may be obtained from a solution of the linearized equations by placing the solution on Mach linees computed from the solution. This is a natural extension of Whitham’s method for the external flow problem. However, it does not suffice to use the ordinary solution of the linearized equations as this possesses singularities. It is necessary to obtain a solution of the linearized equations satisfying boundary conditions in which due allowance has been made for the non-parallelism of the Mach lines. Within the accuracy of the approximation, this solution is found to agree with the ordinary solution away from the wave fronts but differs markedly within them. A simple method is obtained for converting the singular portions of the ordinary solution into a form valid within wave fronts. The problem, studied by Meyer and Ward, of an expansive discontinuity in the slope of the wall of the duct is discussed and the details of the flow are clarified. It is shown that both the velocity and the velocity gradients are finite on the Mach lines where previous theories predicted singularities. Nevertheless, a shock wave is formed in the reflexion of the expansion wave from the axis of the duct, no matter how small the initial disturbance.