Internal problem ID [6494]
Internal file name [OUTPUT/5742_Sunday_June_05_2022_03_52_23_PM_43449969/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(c).
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+2)*y(x)+diff(y(x),x), 1/x*y(x)-2/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: 0.031 (sec). Leaf size: 120
Order:=8; dsolve(x^3*diff(y(x),x$3)-2*x^2*diff(y(x),x$2)+(x^2+2*x)*diff(y(x),x)-x*y(x)=0,y(x),type='series',x=0);
\[ y \left (x \right ) = c_{1} x^{3} \left (1-\frac {1}{4} x +\frac {1}{40} x^{2}-\frac {1}{720} x^{3}+\frac {1}{20160} x^{4}-\frac {1}{806400} x^{5}+\frac {1}{43545600} x^{6}-\frac {1}{3048192000} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} x^{2} \left (\ln \left (x \right ) \left (\left (-240\right ) x +60 x^{2}-6 x^{3}+\frac {1}{3} x^{4}-\frac {1}{84} x^{5}+\frac {1}{3360} x^{6}-\frac {1}{181440} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (720-908 x +152 x^{2}-11 x^{3}+\frac {4}{9} x^{4}-\frac {79}{7056} x^{5}+\frac {517}{2822400} x^{6}-\frac {851}{457228800} x^{7}+\operatorname {O}\left (x^{8}\right )\right )\right )+c_{3} \left (2 \ln \left (x \right ) \left (x^{3}-\frac {1}{4} x^{4}+\frac {1}{40} x^{5}-\frac {1}{720} x^{6}+\frac {1}{20160} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (-24-12 x -6 x^{2}+\frac {5}{8} x^{4}-\frac {39}{400} x^{5}+\frac {49}{7200} x^{6}-\frac {199}{705600} x^{7}+\operatorname {O}\left (x^{8}\right )\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.664 (sec). Leaf size: 186
AsymptoticDSolveValue[x^3*y'''[x]-2*x^2*y''[x]+(x^2+2*x)*y'[x]-x*y[x]==0,y[x],{x,0,7}]
\[ y(x)\to c_1 \left (\frac {\left (x^3-18 x^2+180 x-720\right ) x^3 \log (x)}{4320}+\frac {-167 x^6+2466 x^5-17100 x^4+14400 x^3+129600 x^2+259200 x+518400}{259200}\right )+c_2 \left (\frac {x^3 \left (x^5-40 x^4+1120 x^3-20160 x^2+201600 x-806400\right ) \log (x)}{2419200}-\frac {x^2 \left (2941 x^6-106720 x^5+2618560 x^4-38666880 x^3+268128000 x^2-225792000 x-2032128000\right )}{2032128000}\right )+c_3 \left (\frac {x^9}{43545600}-\frac {x^8}{806400}+\frac {x^7}{20160}-\frac {x^6}{720}+\frac {x^5}{40}-\frac {x^4}{4}+x^3\right ) \]