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First Order Equations
Published in Vladimir A. Dobrushkin, Applied Differential Equations with Boundary Value Problems, 2017
(e) e-ydx + (x/y)e-ydy = 0; (f) (3x2y - x-2y5)dx = (x3 - 3x-1y4) dy; (g) xdx = (x2y + y3) dy; (h) (yxsecy-tany)dx=(x-secylnx) $ (\frac{y}{x} {\text{sec }}y - {\text{tan }}y)dx = (x - {\text{sec }}y {\text{ln }}x) $ dy.Use an integrating factor to find the general solution of the given differential equations.
First-order differential equations
Published in C.W. Evans, Engineering Mathematics, 2019
The linear equation is an example of a general class of differential equations which can be solved by means of a device known as an integrating factor. An integrating factor is an expression which when multiplied through the equation makes it easy to integrate. Suppose in the case which we are considering ((dy/dx) + Py = Q) it is possible to multiply throughout by some expression l and thereby express the equation in the form ddx(Iy)=IQ
First Order Equations
Published in Vladimir A. Dobrushkin, Applied Differential Equations, 2022
by Theorem 2.3, page 74. In general, we don't know how to solve the partial differential equation (2.4.4) other than transfer it to the ordinary differential equation (2.4.1). In principle, the integrating factor method is a powerful tool for solving differential equations, but in practice, it may be very difficult, perhaps impossible to implement. Usually, an integrating factor can be found only under several restrictions imposed on M and N. In this section, we will consider two simple classes when μ is a function of one variable only.
Lie Group Analysis of the Integral Equations Related to Neutron Slowing-Down Theory
Published in Nuclear Science and Engineering, 2022
Patrick O’Rourke, Scott Ramsey, Brian Temple
Although this seems like a long-winded way of solving the ODE [Eq. (41)], we emphasize that this is a more fundamental process of doing so. It also provides insight into what the typical integrating factor method is actually doing: transforming the ODE into an equivalent ODE that is separable. Thus the LIFM should be thought of as a higher-level realization of the traditional integrating factor as it exploits the symmetry properties of the ODE to solve it. In fact, the main lesson from this subsection is that a symmetry group for a first-order inhomogeneous ODE is always given by the transformation group , which corresponds to a translation in the solution space .