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.


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.

  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.