Internal problem ID [6495]
Internal file name [OUTPUT/5743_Sunday_June_05_2022_03_52_25_PM_11483593/index.tex]
Book: Differential Equations: Theory, Technique, and Practice by George Simmons, Steven
Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section: Chapter 4. Power Series Solutions and Special Functions. Problems for review and
discovert. (A) Drill Exercises . Page 194
Problem number: 3(d).
ODE order: 3.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[[_3rd_order, _with_linear_symmetries]]
Unable to solve or complete the solution.
Unable to parse ODE.
Maple trace
`Methods for third order ODEs: --- Trying classification methods --- trying a quadrature checking if the LODE has constant coefficients checking if the LODE is of Euler type trying high order exact linear fully integrable trying to convert to a linear ODE with constant coefficients trying differential order: 3; missing the dependent variable Equation is the LCLM of -1/x*y(x)+diff(y(x),x), -(2*x+1)/x^2*y(x)+2*(x-1)/x*diff(y(x),x)+diff(diff(y(x),x),x) trying differential order: 1; missing the dependent variable checking if the LODE is of Euler type <- LODE of Euler type successful Euler equation successful trying a quadrature checking if the LODE has constant coefficients checking if the LODE is of Euler type trying a symmetry of the form [xi=0, eta=F(x)] checking if the LODE is missing y -> Trying a Liouvillian solution using Kovacics algorithm <- No Liouvillian solutions exists -> Trying a solution in terms of special functions: -> Bessel <- Bessel successful <- special function solution successful <- solving the LCLM ode successful `
✓ Solution by Maple
Time used: 0.032 (sec). Leaf size: 1506
Order:=8; dsolve(x^3*diff(y(x),x$3)+(2*x^3-x^2)*diff(y(x),x$2)-x*diff(y(x),x)+y(x)=0,y(x),type='series',x=0);
\[ y \left (x \right ) = c_{3} x \left (1+\operatorname {O}\left (x^{8}\right )\right )+c_{2} x^{\frac {3}{2}-\frac {\sqrt {13}}{2}} \left (1-x +\frac {-3+\sqrt {13}}{-4+2 \sqrt {13}} x^{2}+\frac {5-\sqrt {13}}{6 \sqrt {13}-12} x^{3}+\frac {1}{24} \frac {\left (-5+\sqrt {13}\right ) \left (-7+\sqrt {13}\right )}{\left (-2+\sqrt {13}\right ) \left (-4+\sqrt {13}\right )} x^{4}+\frac {1}{30} \frac {-19+4 \sqrt {13}}{\left (-2+\sqrt {13}\right ) \left (-4+\sqrt {13}\right )} x^{5}+\frac {1}{20} \frac {-29+7 \sqrt {13}}{\left (-2+\sqrt {13}\right ) \left (-4+\sqrt {13}\right ) \left (-6+\sqrt {13}\right )} x^{6}+\frac {-\frac {117}{35}+\frac {6 \sqrt {13}}{7}}{\left (-2+\sqrt {13}\right ) \left (-4+\sqrt {13}\right ) \left (-6+\sqrt {13}\right ) \left (-7+\sqrt {13}\right )} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_{1} x^{\frac {3}{2}+\frac {\sqrt {13}}{2}} \left (1-x +\frac {3+\sqrt {13}}{4+2 \sqrt {13}} x^{2}+\frac {-5-\sqrt {13}}{6 \sqrt {13}+12} x^{3}+\frac {1}{24} \frac {\left (5+\sqrt {13}\right ) \left (7+\sqrt {13}\right )}{\left (2+\sqrt {13}\right ) \left (4+\sqrt {13}\right )} x^{4}-\frac {1}{30} \frac {19+4 \sqrt {13}}{\left (2+\sqrt {13}\right ) \left (4+\sqrt {13}\right )} x^{5}+\frac {1}{20} \frac {29+7 \sqrt {13}}{\left (2+\sqrt {13}\right ) \left (4+\sqrt {13}\right ) \left (6+\sqrt {13}\right )} x^{6}+\frac {-\frac {117}{35}-\frac {6 \sqrt {13}}{7}}{\left (2+\sqrt {13}\right ) \left (4+\sqrt {13}\right ) \left (6+\sqrt {13}\right ) \left (7+\sqrt {13}\right )} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.24 (sec). Leaf size: 310
AsymptoticDSolveValue[x^3*y'''[x]+(2*x^3-x^2)*y''[x]-y'[x]+y[x]==0,y[x],{x,0,7}]
\[ y(x)\to c_1 \left (\frac {99473 x^7}{1008}+\frac {1043 x^6}{144}+\frac {19 x^5}{24}+\frac {11 x^4}{24}-\frac {x^3}{6}+\frac {x^2}{2}+x+1\right )+c_2 e^{-\frac {2}{\sqrt {x}}} \left (-\frac {279112936065458899252220570230691 x^{13/2}}{160251477454333302276096}-\frac {2430057902534044595693470483 x^{11/2}}{100317681699677798400}-\frac {1545013796231079344731 x^{9/2}}{3562417673994240}-\frac {2005991558758787 x^{7/2}}{193273528320}-\frac {43999069453 x^{5/2}}{125829120}-\frac {438565 x^{3/2}}{24576}+\frac {14436319972596450047835320516938615783 x^7}{897408273744266492746137600}+\frac {3840864007433053956366665361751 x^6}{19260994886338137292800}+\frac {1786308115320202497636167 x^5}{569986827839078400}+\frac {319234145332261451 x^4}{4947802324992}+\frac {21959100963217 x^3}{12079595520}+\frac {117706529 x^2}{1572864}+\frac {2353 x}{512}-\frac {29 \sqrt {x}}{16}+1\right ) x^{11/4}+c_3 e^{\frac {2}{\sqrt {x}}} \left (\frac {279112936065458899252220570230691 x^{13/2}}{160251477454333302276096}+\frac {2430057902534044595693470483 x^{11/2}}{100317681699677798400}+\frac {1545013796231079344731 x^{9/2}}{3562417673994240}+\frac {2005991558758787 x^{7/2}}{193273528320}+\frac {43999069453 x^{5/2}}{125829120}+\frac {438565 x^{3/2}}{24576}+\frac {14436319972596450047835320516938615783 x^7}{897408273744266492746137600}+\frac {3840864007433053956366665361751 x^6}{19260994886338137292800}+\frac {1786308115320202497636167 x^5}{569986827839078400}+\frac {319234145332261451 x^4}{4947802324992}+\frac {21959100963217 x^3}{12079595520}+\frac {117706529 x^2}{1572864}+\frac {2353 x}{512}+\frac {29 \sqrt {x}}{16}+1\right ) x^{11/4} \]