Solve the Differential Equation: (1 - x^2) y'' - (4x) y' + (6) y = 0

In summary, the problem is to solve the differential equation (1 + x^2)y'' - (4x)y' + 6y = 0 with initial condition x_0 = 0. The solution is worked out in a similar fashion to the previous problem (1 - x^2)y'' - (4x)y' + 6y = 0, with the main difference being the sign in front of the x^2 term. The final solution is not shown, but it can be found by following the same steps as the previous problem.
  • #1
VinnyCee
489
0
Here is the problem (number 9 in 5.2 of Boyce, DiPrima 8th Edition Book):

[tex](1 - x^2)\,y'' - (4x)\,y' + (6)\,y = 0,\,x_0 = 0[/tex]

Here is what I have so far:

[tex]\sum_{n = 0}^{\infty}\,(n + 2)\,(n + 1)\,a_{n + 2}\,x^n - \sum_{n = 2}^{\infty}\,n(n - 1)\,a_n\,x^n - 4\,\sum_{n = 1}^{\infty}\,n\,a_n\,x^n + 6\,\sum_{n = 0}^{\infty}\,a_n\,x^n = 0[/tex]

Now, I took out the first two terms (n = 0 and n = 1) of the first sum to make it's index go to n = 2 in order to add all the sums together, right?

[tex]2\,a_2 + 6\,a_3\,x[/tex]

Continuing in that fashion to add the sums, i get this:

[tex]2\,a_2\,+\,6\,a_3\,x\,-\,4\,a_1\,x\,+\,6\,a_0\,+\,6\,a_1\,x\,+\,\sum_{n = 2}^{\infty}\,\left[(n\,+\,2)\,(n\,+\,1)\,a_{n\,+\,2}\,-\,n\,(n\,-\,1)\,a_n\,-\,4\,n\,a_n\,+\,6\,a_n\right]\,x^n\,=\,0[/tex]

This is where I am stuck. I am not sure if I pulled out those factors correctly or what, please help. Thank you.
 
Physics news on Phys.org
  • #2
Whoops!

I copied the problem wrong!

It is supposed to be:

[tex](1 + x^2)\,y'' - (4x)\,y' + (6)\,y = 0,\,x_0 = 0[/tex]

And NOT:

[tex](1 - x^2)\,y'' - (4x)\,y' + (6)\,y = 0,\,x_0 = 0[/tex]

One little minus sign!

The whole solution is worked out as the last problem http://tutorial.math.lamar.edu/AllBrowsers/3401/SeriesSolutions.asp [Broken]!
 
Last edited by a moderator:
  • #3


To solve this differential equation, we can use the method of power series. First, we can rewrite the given equation as:

y'' - \frac{4x}{1-x^2}y' + \frac{6}{1-x^2}y = 0

Next, we can substitute the power series representation of y into the equation:

y = \sum_{n=0}^\infty a_nx^n

y' = \sum_{n=0}^\infty (n+1)a_{n+1}x^n

y'' = \sum_{n=0}^\infty (n+2)(n+1)a_{n+2}x^n

Substituting these into the original equation, we get:

\sum_{n=0}^\infty (n+2)(n+1)a_{n+2}x^n - \frac{4x}{1-x^2}\sum_{n=0}^\infty (n+1)a_{n+1}x^n + \frac{6}{1-x^2}\sum_{n=0}^\infty a_nx^n = 0

We can simplify this to:

\sum_{n=0}^\infty ((n+2)(n+1)a_{n+2} - 4(n+1)a_{n+1} + 6a_n)x^n = 0

Since this holds for all values of x, we can equate the coefficients of each power of x to 0. This gives us the following recurrence relation for the coefficients:

(n+2)(n+1)a_{n+2} - 4(n+1)a_{n+1} + 6a_n = 0

We can solve for the coefficients using this recurrence relation. Starting with n = 0, we can solve for a_0. Then, using a_0, we can solve for a_1. Continuing in this manner, we can find all the coefficients a_n.

Finally, we can substitute the values of a_n into the power series representation of y to get the solution to the differential equation.
 

1. What is a differential equation?

A differential equation is a mathematical equation that describes the relationship between a function and its derivatives. It is used to model many real-world phenomena in fields such as physics, engineering, and economics.

2. How do you solve a differential equation?

To solve a differential equation, you need to find a function that satisfies the equation. This can be done through various methods such as separation of variables, substitution, or using integration techniques.

3. What is the order of this differential equation?

The order of a differential equation is the highest derivative present in the equation. In this case, the order is 2 because the equation contains the second derivative of y.

4. Why is it important to solve differential equations?

Differential equations are important because they allow us to model and understand complex systems and phenomena in various fields. They also have practical applications in engineering, physics, and other scientific fields.

5. Can this differential equation be solved analytically?

Yes, this differential equation can be solved analytically using various methods such as separation of variables or substitution. However, in some cases, it may be difficult or impossible to find an exact solution and numerical methods may be used instead.

Similar threads

  • Calculus and Beyond Homework Help
Replies
8
Views
925
  • Differential Equations
Replies
7
Views
308
  • Introductory Physics Homework Help
Replies
8
Views
359
  • Calculus and Beyond Homework Help
Replies
2
Views
640
Replies
3
Views
530
  • Calculus and Beyond Homework Help
Replies
6
Views
224
  • Differential Equations
Replies
1
Views
1K
Replies
6
Views
477
  • Introductory Physics Homework Help
Replies
10
Views
1K
  • Introductory Physics Homework Help
Replies
2
Views
919
Back
Top