Internal problem ID [13671]
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 (e).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {x^{2} y^{\prime \prime }-y^{\prime } x +\frac {y}{1-x}=0} \] With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 47
Order:=6; dsolve(x^2*diff(y(x),x$2)-x*diff(y(x),x)+1/(1-x)*y(x)=0,y(x),type='series',x=0);
\[ y \left (x \right ) = \left (\left (\ln \left (x \right ) c_{2} +c_{1} \right ) \left (1-x +\operatorname {O}\left (x^{6}\right )\right )+\left (2 x -\frac {1}{2} x^{2}-\frac {1}{6} x^{3}-\frac {1}{12} x^{4}-\frac {1}{20} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \right ) x \]
✓ Solution by Mathematica
Time used: 0.006 (sec). Leaf size: 60
AsymptoticDSolveValue[x^2*y''[x]-x*y'[x]+1/(1-x)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_2 \left (x \left (-\frac {x^5}{20}-\frac {x^4}{12}-\frac {x^3}{6}-\frac {x^2}{2}+2 x\right )+(1-x) x \log (x)\right )+c_1 (1-x) x \]