Solving 2nd Order Homogeneous Equations with Non-Constant Coefficients

[beetle2]

Hi Guys,

I know how to find the solution to a 2nd order homogeneous with constant coefficients but how do you solve one with a non constant
ie

x^2y''+2xy' ... etc = 0

Is there a general solution formula for these types of problems?

My book seems to jump from 2nd order homogeneous with constant coefficients straight to
2nd order non homogeneous with undetermined coefficients.
any help greatly appreciated.

 

[ross_tang]

Hello, beetle2. It didn't mentioned 2nd order with variable coefficients, since there is no general method in solving that. However, what you can do is by guessing the first solution. Then you can find the 2nd one using various method, such as:

http://www.voofie.com/content/84/solving-linear-non-homogeneous-ordinary-differential-equation-with-variable-coefficients-with-operat/"

At last, I don't really like undetermined coefficients method, which is highly restricted in the class of function you can solve. You may want to have a look at:

http://www.voofie.com/content/6/introduction-to-differential-equation-and-solving-linear-differential-equations-using-operator-metho/"

Lastly, for various tutorial, paper, articles, discussion in Ordinary differential equations, you may refer to:

http://www.voofie.com/concept/Ordinary_differential_equation/"

 

[HallsofIvy]

The specific example you give, with the coefficient of each derivative being a power of x, with power equal to the degree of the derivative, $ax^2y''+ bxy+ cy= 0$ is an "equi-potential" or "Euler-type" equation. "Trying" a solution of the form $x^r$, in the same way you "try" a solution of the form $e^{rx}$ for equations with constant coefficients.

That works because the substitution t= ln(x) reduces an "Euler-type" equation with independent variable x to a "constant coefficients" equation with independent variable t.

For more general linear equations with variable coefficients, you typically have to try power series solutions. Let $y= \sum a_n x^n$, substitute that and its derivatives into the equation and try to find the coefficients, $a_n$.

 

[aq1q]

hm, are you familiar with the Frobenius method. This is where you let $y= \sum a_n x^{n + r}$

Check it out: http://en.wikipedia.org/wiki/Frobenius_method

 

[blue_raver22]

Hello,

Thank you for your question. Solving 2nd order homogeneous equations with non-constant coefficients can be a bit more challenging than solving those with constant coefficients. However, the general approach is to use the method of undetermined coefficients or the method of variation of parameters. Both of these methods allow you to find a particular solution to the equation, which can then be combined with the general solution of the corresponding homogeneous equation to obtain the complete solution.

In the case of non-constant coefficients, the particular solution can be found by assuming a form for the solution (e.g. exponential, polynomial, trigonometric) and then substituting it into the equation to determine the coefficients. This can be a trial and error process, but it is usually more straightforward when compared to solving non-homogeneous equations.

It is also worth noting that there is no general solution formula for these types of problems. The solution will depend on the specific form of the non-constant coefficients in the equation.

I hope this helps. Please let me know if you have any further questions. Best of luck with your studies!