Internal problem ID [13998]
Book: Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell.
second edition. CRC Press. FL, USA. 2020
Section: Chapter 35. Modified Power series solutions and basic method of Frobenius. Additional
Exercises. page 715
Problem number: 35.4 (L).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {4 x^{2} y^{\prime \prime }+8 x^{2} y^{\prime }+y=0} \] With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 67
Order:=6; dsolve(4*x^2*diff(y(x),x$2)+8*x^2*diff(y(x),x)+y(x)=0,y(x),type='series',x=0);
\[ y \left (x \right ) = \left (\left (c_{1} +c_{2} \ln \left (x \right )\right ) \left (1-x +\frac {3}{4} x^{2}-\frac {5}{12} x^{3}+\frac {35}{192} x^{4}-\frac {21}{320} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (-\frac {1}{4} x^{2}+\frac {1}{4} x^{3}-\frac {19}{128} x^{4}+\frac {25}{384} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \right ) \sqrt {x} \]
✓ Solution by Mathematica
Time used: 0.003 (sec). Leaf size: 125
AsymptoticDSolveValue[4*x^2*y''[x]+8*x^2*y'[x]+y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 \sqrt {x} \left (-\frac {21 x^5}{320}+\frac {35 x^4}{192}-\frac {5 x^3}{12}+\frac {3 x^2}{4}-x+1\right )+c_2 \left (\sqrt {x} \left (\frac {25 x^5}{384}-\frac {19 x^4}{128}+\frac {x^3}{4}-\frac {x^2}{4}\right )+\sqrt {x} \left (-\frac {21 x^5}{320}+\frac {35 x^4}{192}-\frac {5 x^3}{12}+\frac {3 x^2}{4}-x+1\right ) \log (x)\right ) \]