Internal problem ID [13690]
Book: Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell.
second edition. CRC Press. FL, USA. 2020
Section: Chapter 36. The big theorem on the the Frobenius method. Additional Exercises. page
739
Problem number: 36.2 (f).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {x^{2} \left (2 x +1\right ) y^{\prime \prime }+y^{\prime } x +\left (4 x^{3}-4\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 63
Order:=6; dsolve(x^2*(1+2*x)*diff(y(x),x$2)+x*diff(y(x),x)+(4*x^3-4)*y(x)=0,y(x),type='series',x=0);
\[ y \left (x \right ) = \frac {c_{1} x^{4} \left (1-\frac {4}{5} x +\frac {4}{5} x^{2}-\frac {116}{105} x^{3}+\frac {311}{210} x^{4}-\frac {358}{175} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (576 x^{4}-\frac {2304}{5} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (-144-576 x -576 x^{2}-192 x^{3}-384 x^{4}+\frac {15744}{25} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right )}{x^{2}} \]
✓ Solution by Mathematica
Time used: 0.044 (sec). Leaf size: 77
AsymptoticDSolveValue[x^2*(1+2*x)*y''[x]+x*y'[x]+(4*x^3-4)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 \left (\frac {19 x^4+4 x^3+12 x^2+12 x+3}{3 x^2}-4 x^2 \log (x)\right )+c_2 \left (\frac {311 x^6}{210}-\frac {116 x^5}{105}+\frac {4 x^4}{5}-\frac {4 x^3}{5}+x^2\right ) \]