20.26.31 problem 25

Internal problem ID [4056]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 11, Series Solutions to Linear Differential Equations. Exercises for 11.5. page 771
Problem number : 25
Date solved : Monday, January 27, 2025 at 08:07:12 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }-x \left (1-x \right ) y^{\prime }+\left (1-x \right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.016 (sec). Leaf size: 36

Order:=6; 
dsolve(x^2*diff(y(x),x$2)-x*(1-x)*diff(y(x),x)+(1-x)*y(x)=0,y(x),type='series',x=0);
 

\[ y \left (x \right ) = x \left (\left (1+\operatorname {O}\left (x^{6}\right )\right ) \left (c_{2} \ln \left (x \right )+c_{1} \right )+\left (-x +\frac {1}{4} x^{2}-\frac {1}{18} x^{3}+\frac {1}{96} x^{4}-\frac {1}{600} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \right ) \]

✓ Solution by Mathematica

Time used: 0.006 (sec). Leaf size: 50

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

\[ y(x)\to c_2 \left (x \left (-\frac {x^5}{600}+\frac {x^4}{96}-\frac {x^3}{18}+\frac {x^2}{4}-x\right )+x \log (x)\right )+c_1 x \]