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asrxiiviii
asrxiiviii
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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.

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 $c$. 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). 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.

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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asrxiiviii
asrxiiviii
  • 739
  • 4
  • 10

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 $c$. 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). 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.