Internal problem ID [13669]
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 (c).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {x^{2} y^{\prime \prime }+y^{\prime } x +\left (4 x -4\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 61
Order:=6; dsolve(x^2*diff(y(x),x$2)+x*diff(y(x),x)+(4*x-4)*y(x)=0,y(x),type='series',x=0);
\[ y \left (x \right ) = \frac {c_{1} x^{4} \left (1-\frac {4}{5} x +\frac {4}{15} x^{2}-\frac {16}{315} x^{3}+\frac {2}{315} x^{4}-\frac {8}{14175} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (256 x^{4}-\frac {1024}{5} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (-144-192 x -192 x^{2}-256 x^{3}+\frac {6144}{25} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right )}{x^{2}} \]
✓ Solution by Mathematica
Time used: 0.021 (sec). Leaf size: 79
AsymptoticDSolveValue[x^2*y''[x]+x*y'[x]+(4*x-4)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 \left (\frac {4 x^4+16 x^3+12 x^2+12 x+9}{9 x^2}-\frac {16}{9} x^2 \log (x)\right )+c_2 \left (\frac {2 x^6}{315}-\frac {16 x^5}{315}+\frac {4 x^4}{15}-\frac {4 x^3}{5}+x^2\right ) \]