# Looking for references on higher order homogenous differential equations and a particular equation I am trying to solve

Most of the theory I know (and found, after some significant amount of searching) on homogenous higher order differential equations (third order onwards) assume constant coefficients: that is, it is assumed that the equation is of the form $$y'''(x) + ay''(x) + by'(x) + cy(x) = 0$$ for some constants (say real numbers) $$a$$, $$b$$ and $$c$$ (which from what I learnt is called the method of undetermined coefficients). I was however interested in knowing if there is a method yielding the general solution, when instead of constants $$a, b, c, d$$, we have (say smooth) real functions $$a(x), b(x), c(x), d(x)$$, that is when our equation is of the form
$$a(x) y'''(x) + b(x) y''(x) + c(x) y'(x) + d(x) y(x) = 0$$ for $$\mathcal C^\infty$$ functions $$a, b, c, d: \mathbb R \rightarrow \mathbb R$$. (I rewrite without normalizing the first coefficient for a reason that shall be clear soon...) More precisely, the case I am interested in is when the four functions $$a, b, c, d$$ are polynomials and even more specifically, when $$a(x):= x^2(x^2 - 34x + 1), \hspace{2mm} b(x):= 3x(2x^2-51x+1), \hspace{2mm} c(x):=7x^2-112x+1, \hspace{2mm} d(x):= x-5$$ Of course I can find some solutions by forcing it down to a second order differential equation by assuming a solution of the form $$y(x):= \alpha(x) u(x) + \beta(x) u'(x) + \gamma(x) u''(x)$$ for functions $$\alpha, \beta, \gamma$$ obtained by substituting back into the original DE. But I am not sure if this will yield all solutions - I will have to show that every solution if of the aforementioned form: I feel like I have an intuitive argument for this which seems to work but it is difficult to make it rigorous enough, plus I fear pathological counterexamples.

Like I said, I tried looking for general theory on this, but I haven't found this treatment, or any general method for the case when $$a, \cdots , d$$ are polynomials (and its analogues for higher order homogeneous DE's with polynomial coefficients - maybe I am not searching with the right terminology; I don't specialize in differential equations). So, besides the above question (on how to find the general solution in my case), I also wanted to know of any references which provide such a treatment. I would really appreciate any suggestions or references. Thank you.

dsolve(x^2*(x^2 - 34*x + 1)*diff(y(x), x, x, x) + 3*x*(2*x^2 - 51*x + 1)*diff(y(x), x, x) + (7*x^2 - 112*x + 1)*diff(y(x), x) + (x - 5)*y(x)=0);

performs $$y \left( x \right) ={\it \_C1}\, \left( -x+17+12\,\sqrt {2} \right) \left( {\it HeunG} \left( -576-408\,\sqrt {2},-42-30\,\sqrt {2},1,1,1 /2,1,- \left( 17+12\,\sqrt {2} \right) \left( x-17+12\,\sqrt {2} \right) \right) \right) ^{2}+{\it \_C2}\, \left( {x}^{2}-34\,x+1 \right) \left( {\it HeunG} \left( -576-408\,\sqrt {2},-234\,\sqrt {2 }-{\frac{1317}{4}},3/2,3/2,3/2,1,- \left( 17+12\,\sqrt {2} \right) \left( x-17+12\,\sqrt {2} \right) \right) \right) ^{2}+{\it \_C3}\, \sqrt {-x+17+12\,\sqrt {2}}{\it HeunG} \left( -576-408\,\sqrt {2},-42- 30\,\sqrt {2},1,1,1/2,1,- \left( 17+12\,\sqrt {2} \right) \left( x-17 +12\,\sqrt {2} \right) \right) \sqrt {{x}^{2}-34\,x+1}{\it HeunG} \left( -576-408\,\sqrt {2},-234\,\sqrt {2}-{\frac{1317}{4}},3/2,3/2,3 /2,1,- \left( 17+12\,\sqrt {2} \right) \left( x-17+12\,\sqrt {2} \right) \right) .$$