Internal problem ID [11925]
Book: Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi.
2004.
Section: Chapter 6, Series solutions of linear differential equations. Section 6.2 (Frobenius).
Exercises page 251
Problem number: 24.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_Emden, _Fowler]]
\[ \boxed {2 x y^{\prime \prime }+6 y^{\prime }+y=0} \] With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 62
Order:=6; dsolve(2*x*diff(y(x),x$2)+6*diff(y(x),x)+y(x)=0,y(x),type='series',x=0);
\[ y \left (x \right ) = \frac {c_{1} \left (1-\frac {1}{6} x +\frac {1}{96} x^{2}-\frac {1}{2880} x^{3}+\frac {1}{138240} x^{4}-\frac {1}{9676800} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) x^{2}+c_{2} \left (\ln \left (x \right ) \left (\frac {1}{4} x^{2}-\frac {1}{24} x^{3}+\frac {1}{384} x^{4}-\frac {1}{11520} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (-2-x +\frac {1}{18} x^{3}-\frac {25}{4608} x^{4}+\frac {157}{691200} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right )}{x^{2}} \]
✓ Solution by Mathematica
Time used: 0.018 (sec). Leaf size: 80
AsymptoticDSolveValue[2*x*y''[x]+6*y'[x]+y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_2 \left (\frac {x^4}{138240}-\frac {x^3}{2880}+\frac {x^2}{96}-\frac {x}{6}+1\right )+c_1 \left (\frac {31 x^4-352 x^3+576 x^2+4608 x+9216}{9216 x^2}-\frac {1}{768} \left (x^2-16 x+96\right ) \log (x)\right ) \]