Then,
P.D. Another method to solve the equation is to try solutions of the form .
ok in this problem im supposed to use the substitution x = e^t to transform the given Cauchy-Euler equation to a differential equation with constant coefficients.
x^2y'' +9xy' -20y=0
im not sure how to start this problem
maybe dy/dx= (dy/dt)(dt/dx) = (dy/dt)e^t
so then could i say d2y/dx^2 = (e^t)d2y/dt^2<---- ?
maybe then i could substitute back into the equation? i dunnno..
any help would be appreciated thanks in advance.
The problem with simply taking , as both Fernando Revilla and Ackbeet suggest, is that it not clear what to do with "special cases". For example, consider the differential equation . Taking the equation reduces to which gives the characteristic equation . Okay, one solution is but what is the other, independent, solution we need? And what should we do about that right side, x+ ln(x)?
If we know that the substitution changes the Euler-Cauchy equation to the constant-coefficients equation with the same characteristic equation, we know that the solution to the assosciated homogeneous equation, in terms of t, is , which tells us that the corresponding solution to the original equation is . Also, the right hand side, in terms of t, is , which, by "undetermined coefficients" has particular solution so that the entire general solution to the original equation is
.
Of course, we could learn variations of all those rules for Cauchy-Euler equations but it is much simpler to know how to convert such an equation to a constant-coefficients equation.