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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.

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The command of Maple

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

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  • $\begingroup$ Thanks for the solution in this case. I am just looking at what the Heun General Differential Equation is but I presume this is the general solution. I was also wondering if the method I proposed above gives all the general solutions, and also for references on how to find general solutions to homogenous higher order ODE's with polynomial coefficients. If you happen to know some information about these, would you would like to share the same? $\endgroup$
    – asrxiiviii
    Oct 15 '20 at 13:17
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While @user64494's answer is strictly related to your particular case, I will try to give a general overview of the situation. A homogeneous linear differential equation of the first order can be solved by quadrature (that is, it can be solved via the use of integrals). For higher order equations, we find a situation which is similar to the one of algebraic equations. In fact, using Galois theory we can determine whether an algebraic equation can be solved by radicals or not. Similarly, for higher order homogeneous linear differential equations with variable coefficients, it is possible to develop a theory (now called differential Galois theory) which allows us to understand whether an equation can be solved by quadrature (i.e. via integrals) or not. The first developments of differential Galois theory are due to Picard and Vessiot. A good reference for these topics is van der Put, M; Singer, M. F. (2003). Galois Theory of linear differential equations. Grundlehren der Mathematischen Wissenschaften. Berlin, New York: Springer-Verlang.

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