20.25.1 problem 1

Internal problem ID [4006]
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 11.4. page 758
Problem number : 1
Date solved : Monday, January 27, 2025 at 08:06:04 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.002 (sec). Leaf size: 49

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

\[ y \left (x \right ) = \left (1-\frac {1}{6} x^{3}+\frac {1}{24} x^{4}+\frac {1}{60} x^{5}\right ) y \left (0\right )+\left (x -\frac {1}{2} x^{2}-\frac {1}{12} x^{4}+\frac {1}{24} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.002 (sec). Leaf size: 56

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

\[ y(x)\to c_1 \left (\frac {x^5}{60}+\frac {x^4}{24}-\frac {x^3}{6}+1\right )+c_2 \left (\frac {x^5}{24}-\frac {x^4}{12}-\frac {x^2}{2}+x\right ) \]