20.24.16 problem Problem 17

Internal problem ID [4001]
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 11.2. page 739
Problem number : Problem 17
Date solved : Monday, January 27, 2025 at 08:06:00 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.016 (sec). Leaf size: 40

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

\[ y \left (x \right ) = \left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1+\frac {1}{4} x^{2}+\frac {1}{18} x^{3}+\frac {5}{192} x^{4}+\frac {23}{3600} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (x +\frac {11}{108} x^{3}+\frac {11}{1152} x^{4}+\frac {883}{216000} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \]

✓ Solution by Mathematica

Time used: 0.005 (sec). Leaf size: 96

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

\[ y(x)\to c_1 \left (\frac {23 x^5}{3600}+\frac {5 x^4}{192}+\frac {x^3}{18}+\frac {x^2}{4}+1\right )+c_2 \left (\frac {883 x^5}{216000}+\frac {11 x^4}{1152}+\frac {11 x^3}{108}+\left (\frac {23 x^5}{3600}+\frac {5 x^4}{192}+\frac {x^3}{18}+\frac {x^2}{4}+1\right ) \log (x)+x\right ) \]