This discussion clarifies the appropriate methods for solving ordinary differential equations (ODEs). The consensus is to use the trial function y=e^mx for ODEs with constant coefficients, while y=x^m is suitable for Cauchy-Euler equations. The derivatives of e^{ax} yield similar forms, making them effective for linear differential equations with constant coefficients. In contrast, the derivatives of x^a produce varying powers, which align with linear differential equations that have powers of x as coefficients.
PREREQUISITESStudents and educators in mathematics, particularly those focused on differential equations, as well as anyone seeking clarity on the application of different trial functions in solving ODEs.
!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 !