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Can anybody give any hints how to solve any kind of particular solution involving at least one arbitrary constant of this ode: $p^2y^{\prime \prime \prime}+ yy^{\prime}+y+ax+b=0$,

where $a$,$b$,$p$ are constants.

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Ansatz: $y=cx+d$, $y'=c$, $y'''=0$. $$0=p^2y'''+yy'+y+ax+b=(cx+d)c+cx+d+ax+b= (c^2+c+a)x+(cd+d+b).$$ Conclude: $y$ solves your ODE if: $$c=\frac{-1\pm\sqrt{1-4a}}{2},$$ $$d=-\frac{b}{c+1}.$$

So there's two solutions. I do not know if there are others.

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  • $\begingroup$ and if $y(x)$ is an another function? $\endgroup$ – Dr. Sonnhard Graubner Jan 20 '18 at 14:29
  • $\begingroup$ There may be more solutions for $y(x)$. But I have found two of them. $\endgroup$ – Alex S Jan 20 '18 at 14:33
  • $\begingroup$ Thanks for help. But in my problem, I need atleast one arbitrary constant, but in this case the arbitrary constants $c$ and $d$ are not arbitrary, since they are particular functions of $a$ and $b$. Can you please give some suggestion on that? $\endgroup$ – Subhankar Sil Jan 20 '18 at 14:38
  • $\begingroup$ That is a much tougher problem, and I don't know how to answer it. Good luck. Shall I delete my answer? $\endgroup$ – Alex S Jan 20 '18 at 14:43
  • $\begingroup$ No, please let it be. Thanks for your kind help. Will look forward to hear from you. $\endgroup$ – Subhankar Sil Jan 20 '18 at 14:46

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