# Solve DE: m, h, E, Z, t, k, a (1-4)

• A
• OneByBane
In summary, the given differential equation is a second order equation that can be rewritten as the Whittaker equation. It can also be written in terms of the Coulomb wave functions or confluent hypergeometric functions. To solve it, a particular solution must be found to determine the general solution.
OneByBane
I am currently trying to solve this differential equation:

r2/F(r) d2F(r)/dr2 + 2mr2/h2(E + Zt2/kr) - a2 = 0

Wher m, h, E, Z, t and k are other variables and 'a' can have values 1, 2, 3, 4... (Whole numbers)

I have come across this while solving a problem in physics and have no clue if this even has a solution.
Any help will be appreciated greatly.

$$\frac{r^2}{f(r)} \frac{d^2 f(r)}{dr^2} + \frac{2mr^2}{h^2}\left(E + \frac{zt^2}{kr}\right) -a^2 = 0$$
Then you can rewrite to
$$\left( \frac{d^2}{dr^2} + \frac{2mE}{h^2} + \frac{2mzt^2}{h^2 kr} - \frac{a^2}{r^2} \right) f(r) = 0$$
Which is pretty close to the Whittaker equation ( https://en.wikipedia.org/wiki/Whittaker_function ).

Ssnow
This is a second order differential equation, you can rewrite it as:

##\frac{d^2}{dr^2}F(r)+ \left[\frac{2m}{h^2}\left(E-\frac{h^2a^2}{2mr^2} + \frac{Zt^2}{kr}\right)\right]F(r) =0##

calling ##a(r)=\frac{2m}{h^2}\left(E-\frac{h^2a^2}{2mr^2} + \frac{Zt^2}{kr}\right)## we have that

##\frac{d^2}{dr^2}F(r)+ a(r)\cdot F(r) =0##

to solve this DE you must find a particular solution in order to find the general ...

The solutions to this equation are called Coulomb wave functions. You can also write the solution in terms of confluent hypergeometric functions.

## What is a differential equation?

A differential equation is a mathematical equation that relates a function or a set of functions to its derivatives. It is used to describe the relationship between a changing quantity and its rate of change.

## What does "m, h, E, Z, t, k, a" represent in a differential equation?

In a differential equation, these variables typically represent different physical quantities such as mass (m), height (h), energy (E), impedance (Z), time (t), spring constant (k), and acceleration (a).

## What are the different methods for solving a differential equation?

There are several methods for solving a differential equation, including separation of variables, substitution, integrating factors, and power series. The appropriate method to use depends on the type and complexity of the differential equation.

## Why are differential equations important in science?

Differential equations are important in science because they provide a mathematical framework for modeling and analyzing complex systems and phenomena in fields such as physics, chemistry, biology, and engineering. They are used to describe and predict the behavior of systems that involve varying quantities over time or space.

## Can differential equations be solved analytically?

In some cases, differential equations can be solved analytically, meaning that an explicit solution can be found. However, many differential equations do not have analytical solutions and require numerical methods for approximation.

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