How do I solve second order linear differential equations with functions of x as coef

**s3a** · October 19th 2011, 08:24 AM

The title should have been: "How do I solve second order linear differential equations with functions of x as coefficients of y?"

I'm trying to do the question attached and my homework consists of a whole lot of these so it would be greatly appreciated if someone could highlight the steps in doing this one so I can reproduce it.

Any help would be greatly appreciated!
Thanks in advance!

**HallsofIvy** · October 19th 2011, 09:10 AM

There are a few special cases- in particular, "Cauchy-Euler type equations" have coefficient of each $d^n y/d^n$ a constant times $x^n$ : $a_nx^n\frac{d^ny}{dx^n}+ a_{n-1}x^{n-1}\frac{d^{n-1}y}{dx^{n-1}}+ \cdot\cdot\cdot+ a_1x\frac{df}{dx}+ a_0f= F(x)$ .
(These are also called "equi-potential equations" because the power of x is the same as the order of the derivative.)
The substitution t= ln(x) changes that type of equation to a "constant coefficients" equation.

But the most general method is to look for a power series solution. That is, write $y= \sum_{n=0}^\infty a_nx^n$ . Find the derivatives of that, put them into the equation (If the coefficients are not polynomials, write Taylor series for them), and you typically can get a "recursion relation" for the numbers $a_n$ .

If the coefficient of the highest order derivative is 0 at the point at which you have your initial values, then you may need to use "Frobenius' method": Frobenius Method -- from Wolfram MathWorld
with something of the form
$\sum_{n=0}^\infty a_nx^{n+c}$
where c is a constant, not necessarily positive or an integer.

**chisigma** · October 19th 2011, 11:07 AM

Originally Posted by s3a

The title should have been: "How do I solve second order linear differential equations with functions of x as coefficients of y?"

I'm trying to do the question attached and my homework consists of a whole lot of these so it would be greatly appreciated if someone could highlight the steps in doing this one so I can reproduce it.

Any help would be greatly appreciated!
Thanks in advance!

The non-homogeneous linear ODE is of the type...

$a_{n}\ x^{n}\ \frac{d^{n} y}{d x^{n}} + a_{n-1}\ x^{n-1}\ \frac{d^{n-1} y}{d x^{n-1}} +...+ a_{1}\ x\ \frac{d y}{d x} + a_{0}\ y=0$ (1)

... and a particular solution of it has the form $y(x)=x^{p}$ . The possible values of p are found considering that is...

$\frac{d^{n} y}{d x^{n}} = p\ (p-1)\ ...\ (p-n+1)\ x^{p-n}$ (2)

... so that a common factor $x^{p}$ can be eliminated and the (1) becomes an algebraic equation of order n in p that has as solutions the n possible value of p...

Kind regards

$\chi$ $\sigma$

**HallsofIvy** · October 19th 2011, 01:11 PM

Chi Sigma, that is specifically "Cauchy-Euler type equation". I interpreted this as asking about general linear equations with variable coefficients.

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