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

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!

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

There are a few special cases- in particular, "Cauchy-Euler type equations" have coefficient of each a constant times : .

(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 . 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 .

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

where c is a constant, not necessarily positive or an integer.

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

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

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