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vela said:Rewrite the differential equation slightly to get
[tex]\rho^2u''(\rho)-l(l+1)u(\rho) = 0[/tex]
Then substitute a trial solution of the form [itex]\rho^r[/itex] and then solve for r.
HallsofIvy said:You basic error is trying to apply a method that works for equations with constant coeficients to an equation with variable coefficients. As a result your characteristic equation, which should not involve the idependent variable, is wrong and everything after that is wrong.
A differential equation is a mathematical equation that describes the relationship between a function and its derivatives. It involves the use of derivatives to represent the rate of change of a function, making it a valuable tool in modeling dynamic systems.
The best way to ensure the accuracy of a calculation for a differential equation is to double-check your work and make sure that it satisfies the conditions set by the equation. This includes checking boundary conditions and initial values, as well as verifying that the solution satisfies the equation itself.
Differential equations are important in many fields of science and engineering as they allow us to model and understand complex systems. They are used to predict the behavior of physical systems, such as population growth, chemical reactions, and electrical circuits.
There are several methods for solving differential equations, including separation of variables, variation of parameters, and using a power series. The method used depends on the type of differential equation and the initial conditions given.
The best way to improve your skills in solving differential equations is to practice regularly and familiarize yourself with different types of equations and their corresponding solution methods. You can also seek help from a tutor or online resources to gain a better understanding of the concepts and techniques involved.