Internal problem ID [6492]
Internal file name [OUTPUT/5740_Sunday_June_05_2022_03_52_06_PM_96721303/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(a).
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 trying Louvillian solutions for 3rd order ODEs, imprimitive case -> pFq: Equivalence to the 3F2 or one of its 3 confluent cases under a power @ Moebius <- pFq successful: received ODE is equivalent to the 1F2 ODE, case c = 0 `
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 1219
Order:=8; dsolve(x^3*diff(y(x),x$3)+2*x^2*diff(y(x),x$2)+(x+x^2)*diff(y(x),x)+x*y(x)=0,y(x),type='series',x=0);
\[ y \left (x \right ) = c_{1} \sqrt {x}\, x^{-\frac {i \sqrt {3}}{2}} \left (1+\frac {1}{i \sqrt {3}-1} x -\frac {1}{2+6 i \sqrt {3}} x^{2}-\frac {1}{18} \frac {1}{\left (\frac {\sqrt {3}}{3}+i\right ) \left (-2+i \sqrt {3}\right ) \left (\sqrt {3}+i\right )} x^{3}+\frac {1}{-1728-480 i \sqrt {3}} x^{4}+\frac {1}{3360 i \sqrt {3}+50400} x^{5}-\frac {1}{720} \frac {1}{\left (\sqrt {3}+6 i\right ) \left (\sqrt {3}+5 i\right ) \left (\sqrt {3}+4 i\right ) \left (\sqrt {3}+3 i\right ) \left (2 i+\sqrt {3}\right ) \left (\sqrt {3}+i\right )} x^{6}-\frac {1}{5040} \frac {1}{\left (-2+i \sqrt {3}\right ) \left (\sqrt {3}+7 i\right ) \left (\sqrt {3}+6 i\right ) \left (\sqrt {3}+5 i\right ) \left (\sqrt {3}+4 i\right ) \left (\sqrt {3}+3 i\right ) \left (\sqrt {3}+i\right )} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} \sqrt {x}\, x^{\frac {i \sqrt {3}}{2}} \left (1+\frac {1}{-i \sqrt {3}-1} x +\frac {1}{6 i \sqrt {3}-2} x^{2}+\frac {1}{6} \frac {1}{\left (\sqrt {3}-2 i\right ) \left (-i+\sqrt {3}\right ) \left (i \sqrt {3}+3\right )} x^{3}+\frac {1}{24} \frac {1}{\left (-\sqrt {3}+2 i\right ) \left (i \sqrt {3}+4\right ) \left (i \sqrt {3}+3\right ) \left (-i+\sqrt {3}\right )} x^{4}+\frac {1}{120} \frac {1}{\left (\sqrt {3}-2 i\right ) \left (-i+\sqrt {3}\right ) \left (i \sqrt {3}+5\right ) \left (i \sqrt {3}+4\right ) \left (i \sqrt {3}+3\right )} x^{5}+\frac {1}{720} \frac {1}{\left (-\sqrt {3}+2 i\right ) \left (i \sqrt {3}+6\right ) \left (i \sqrt {3}+5\right ) \left (i \sqrt {3}+4\right ) \left (i \sqrt {3}+3\right ) \left (-i+\sqrt {3}\right )} x^{6}+\frac {1}{5040} \frac {1}{\left (\sqrt {3}-2 i\right ) \left (-i+\sqrt {3}\right ) \left (i \sqrt {3}+7\right ) \left (i \sqrt {3}+6\right ) \left (i \sqrt {3}+5\right ) \left (i \sqrt {3}+4\right ) \left (i \sqrt {3}+3\right )} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_{3} \left (1-x +\frac {1}{3} x^{2}-\frac {1}{21} x^{3}+\frac {1}{273} x^{4}-\frac {1}{5733} x^{5}+\frac {1}{177723} x^{6}-\frac {1}{7642089} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.006 (sec). Leaf size: 3447
AsymptoticDSolveValue[x^3*y'''[x]+2*x^2*y''[x]+(x+x^2)*y'[x]+x*y[x]==0,y[x],{x,0,7}]
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