Solving Differential Equations: A Comprehensive Guide

In summary, the equation x3y′′(x)+(ax3+bx2+cx+d)y(x)=0 requires an high level of special functions. Even with a simpler equation x2y′′(x)+(ax2+bx+c)y(x)=0, the solutions are expressed as a combination of confluent hypergeometric functions (Kummer and Tricomi functions). Nevertheless, in some particular cases, with particular values of parameters a, b, c, d, the solutions can be reduced to functions of lower level.
  • #1
1
0
Hello I seek solutions of differential equation

x3y′′(x)+(ax3+bx2+cx+d)y(x)=0

thank you
 
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  • #2
saidMath said:
Hello I seek solutions of differential equation

x3y′′(x)+(ax3+bx2+cx+d)y(x)=0

thank you

Hi,

That's poorly formatted. But ok you're new. Try your best, if you want to ask other questions, to format it perfectly and to show some effort even if it's minor. So you're looking I think to solve:

[tex] x^3 y''+(ax^3+bx^2+cx+d) y=0[/tex]

I used the math formatting code or Latex to format it. Do a quote on my post to see the code. We can write it as:

[tex]y''+(a+\frac{b}{x}+\frac{c}{x^2}+\frac{d}{x^3})y=0[/tex]

Isn't 0 now an irregular singular point? In fact, it is having rank of 1/2. Until someone comes up with a better approach, let me make this suggestion that you probably won't like at all: Since the rank of the singular point is non-integer, let's for starters, work one with the smallest integer rank of k=1. In that case, we could look at the equation:

[tex]x^4 y''+(x^4+1)y=0[/tex]

Now, look at this link:

http://ocw.mit.edu/courses/mathemat...ngineering-fall-2004/lecture-notes/eight1.pdf

it deals with irregular singular points of rank one. Can we use that paper to solve this equation first? I realize that's taking a bunch of steps backwards but sometimes you have to bust down a lot of walls before you lay the first course. :)

May I ask that we change the title of this to "Solving a DE with an irregular singular point"? That would make it much more interesting.
 
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  • #3
Hi saidMath !
Do you known the hypergeometric functions ?
The equation x3y′′(x)+(ax3+bx2+cx+d)y(x)=0 requieres an high level of special functions.
Even with a simpler equation x2y′′(x)+(ax2+bx+c)y(x)=0, the solutions are expressed as a combination of confluent hypergeometric functions (Kummer and Tricomi functions).
Nevertheless, in some particular cases, with particular values of parameters a, b, c, d, the solutions can be reduced to functions of lower level.
So, the question is : are the parameters a, b, c, d resticted to some particular values ?
If not, do not expect to find the solutions in terms of a combination of a finite number of elementary functions and of usual special functions.
 

1. What is a differential equation?

A differential equation is a mathematical equation that describes the relationship between a function and its derivatives. It involves one or more variables and their rates of change.

2. How is a differential equation different from a regular equation?

A regular equation involves only algebraic operations, while a differential equation involves derivatives and their relationships. Differential equations are used to model real-world phenomena that involve rates of change.

3. What are the applications of differential equations?

Differential equations have wide-ranging applications in various fields such as physics, engineering, economics, and biology. They are used to model and analyze complex systems and predict their behavior over time.

4. What are the different types of differential equations?

There are several types of differential equations, including ordinary differential equations, partial differential equations, and stochastic differential equations. These types differ based on the number of independent variables and the highest order of the derivatives involved.

5. How do you solve a differential equation?

The method for solving a differential equation depends on its type and complexity. Some equations can be solved analytically using techniques such as separation of variables and substitution. For more complex equations, numerical methods such as Euler's method and Runge-Kutta method are used.

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