73.25.19 problem 35.4 (e)

Internal problem ID [15727]
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)
Date solved : Tuesday, January 28, 2025 at 08:06:06 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }-x y^{\prime }+\frac {y}{1-x}&=0 \end{align*}

Using series method with expansion around

\begin{align*} 0 \end{align*}

✓ Solution by Maple

Time used: 0.054 (sec). Leaf size: 38

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);
 

\[ y = x \left (\left (\ln \left (x \right ) c_{2} +c_{1} \right ) \left (1-x +\operatorname {O}\left (x^{6}\right )\right )+\left (2 x -\frac {1}{2} x^{2}-\frac {1}{6} x^{3}-\frac {1}{12} x^{4}-\frac {1}{20} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \right ) \]

✓ Solution by Mathematica

Time used: 0.008 (sec). Leaf size: 60

AsymptoticDSolveValue[x^2*D[y[x],{x,2}]-x*D[y[x],x]+1/(1-x)*y[x]==0,y[x],{x,0,"6"-1}]
 

\[ y(x)\to c_2 \left (x \left (-\frac {x^5}{20}-\frac {x^4}{12}-\frac {x^3}{6}-\frac {x^2}{2}+2 x\right )+(1-x) x \log (x)\right )+c_1 (1-x) x \]