# Pointers for the solution of 2nd order DE with variable coefficient

• crawf_777
In summary, the conversation discusses a 2nd order differential equation with a variable coefficient. The participants suggest using Frobenius' method and finding a series solution to solve the equation. They also mention the use of Kummer and Tricomi functions for the analytical solution.
crawf_777
Hi,

I am looking at the following 2nd order DE with variable coefficient:

y''(x)-(1/x+7)y(x)=0

I would be grateful for any help in regard to methods which may be applied to such an equation.

Many thanks in advance!

C.

Typically such an equation can be solved by looking for a series solution.

The first thing I would do is multiply both sides by x:
$$x\frac{d^2y}{dx^2}- (7x+ 1)y= 0$$

Because of that x in the leading term you will need to use "Frobenius' method"- we look for a solution of the form
$$y(x)= \sum_{n=0}^\infty a_nx^{n+c}$$
where c is not necessarily a positive integer.

Then
$$y'(x)= \sum_{n=0}^\infty (n+c)a_nx^{n+c- 1}$$
$$y''(x)= \sum_{n= 0}^\infty (n+c)(n+c-1)x^{n+c- 2}$$

Put that into the equation to get sums of powers of x equal to 0.

Choose c by looking at the coefficient of $x^0= 1$ and asserting that $a_x$ (n=0) is not 0.

Once you know c, combine like powers and set the coeficients equal to 0. You will get a recursive formula for the coefficients, $a_n$.

Thanks for your help HallsofIvy! Much appreciated.

C.

Hello !

This ODE can be analytically solved. The closed form for the general solutions involves the Kummer function and the Tricomi function ( i.e. the confluent hypergeometric functions)

Hi JJacquelin,
Thanks for your pointer; I got a series solution together ok, though it seems to blow up strangely. Anyway, I would certainly be interested in seeing the analytical form of the solution to said de. I’ll have a look at those functions that you suggested.
Cheers,
C

Hello crawf_777

The analytical solution is :

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Hi JJacquelin,
Thanks for that! Very much appreciated!
C

## 1. What is a pointer in the context of solving 2nd order differential equations?

A pointer is a mathematical tool used to represent the solution of a 2nd order differential equation with variable coefficients. It is a complex number that can be used to find the general solution of the equation.

## 2. How are pointers used to solve 2nd order differential equations with variable coefficients?

Pointers are used in conjunction with the initial conditions of the differential equation to find the general solution. They help to reduce the equation to a simpler form that can be easily solved by using standard techniques.

## 3. What is the significance of variable coefficients in 2nd order differential equations?

The variable coefficients in a 2nd order differential equation represent how the rate of change of the dependent variable is affected by the independent variables. They make the equation more complex and require different methods, such as pointers, to solve.

## 4. Can pointers be used to solve all 2nd order differential equations with variable coefficients?

No, pointers can only be used to solve linear 2nd order differential equations with variable coefficients. Nonlinear equations require different methods, such as numerical or graphical solutions.

## 5. What are some common applications of 2nd order differential equations with variable coefficients?

2nd order differential equations with variable coefficients are commonly used in physics, engineering, and other scientific fields to model real-world phenomena, such as motion, heat flow, and electrical circuits.

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