!I think i figured it out. We're supposed to use y=e^mx when the ode has constant coefficients (a, b, c) and y=x^m for Cauchy-Euler equations, which are ODEs but the terms have have a-sub-n(x^n)(d^n y/dx^n)BvU said:Hello ABearon,
Bit hard to answer without examples. Perhaps you can provide an example for yourself by going the other way: try to compose an exercise where ##x^m## is a useful trial function and another example where ##e^{mx}## is a good choice. Not too hard: simply differentiate and see what kind of DE you can make form the result !
ODEs, or ordinary differential equations, are mathematical equations that involve one or more unknown functions and their derivatives. They are used to model a wide range of phenomena in science and engineering.
ODEs are used to describe the behavior of dynamic systems, such as population growth, chemical reactions, and motion of objects. Therefore, they need to be solved whenever we want to understand the behavior of a system over time.
There are various methods for solving ODEs, including analytical methods, such as separation of variables and integrating factors, and numerical methods, such as Euler's method and Runge-Kutta methods. The choice of method depends on the complexity of the ODE and the desired level of accuracy.
Some common mistakes when solving ODEs include forgetting initial conditions, making algebraic errors, and using incorrect methods. It is important to carefully check the problem and the solution to avoid these mistakes.
Practice and understanding the underlying concepts are key to improving your understanding of solving ODEs. It is also helpful to seek out additional resources, such as textbooks, online tutorials, and practice problems, to supplement your learning.