Internal problem ID [5288]
Book: An introduction to Ordinary Differential Equations. Earl A. Coddington. Dover. NY
1961
Section: Chapter 4. Linear equations with Regular Singular Points. Page 154
Problem number: 1(b).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {3 x^{2} y^{\prime \prime }+x^{6} y^{\prime }+2 x y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.032 (sec). Leaf size: 70
Order:=8; dsolve(3*x^2*diff(y(x),x$2)+x^6*diff(y(x),x)+2*x*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = c_{1} x \left (1-\frac {1}{3} x +\frac {1}{27} x^{2}-\frac {1}{486} x^{3}+\frac {1}{14580} x^{4}-\frac {7291}{656100} x^{5}+\frac {225991}{41334300} x^{6}-\frac {2522341}{3472081200} x^{7}+\mathrm {O}\left (x^{8}\right )\right )+c_{2} \left (\ln \relax (x ) \left (-\frac {2}{3} x +\frac {2}{9} x^{2}-\frac {2}{81} x^{3}+\frac {1}{729} x^{4}-\frac {1}{21870} x^{5}+\frac {7291}{984150} x^{6}-\frac {225991}{62001450} x^{7}+\mathrm {O}\left (x^{8}\right )\right )+\left (1-\frac {1}{3} x^{2}+\frac {14}{243} x^{3}-\frac {35}{8748} x^{4}+\frac {101}{656100} x^{5}+\frac {69199}{14762250} x^{6}+\frac {19882543}{4340101500} x^{7}+\mathrm {O}\left (x^{8}\right )\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.038 (sec). Leaf size: 121
AsymptoticDSolveValue[3*x^2*y''[x]+x^6*y'[x]+2*x*y[x]==0,y[x],{x,0,7}]
\[ y(x)\to c_1 \left (\frac {x \left (7291 x^5-45 x^4+1350 x^3-24300 x^2+218700 x-656100\right ) \log (x)}{984150}+\frac {-80332 x^6+5895 x^5-158625 x^4+2430000 x^3-16402500 x^2+19683000 x+29524500}{29524500}\right )+c_2 \left (\frac {225991 x^7}{41334300}-\frac {7291 x^6}{656100}+\frac {x^5}{14580}-\frac {x^4}{486}+\frac {x^3}{27}-\frac {x^2}{3}+x\right ) \]