20.26.19 problem 13

Internal problem ID [4044]
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 : 13
Date solved : Monday, January 27, 2025 at 08:06:55 AM
CAS classification : [[_Emden, _Fowler]]

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

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.058 (sec). Leaf size: 58

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

\[ y \left (x \right ) = c_{1} x \left (1+\frac {1}{2} x +\frac {1}{12} x^{2}+\frac {1}{144} x^{3}+\frac {1}{2880} x^{4}+\frac {1}{86400} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (x +\frac {1}{2} x^{2}+\frac {1}{12} x^{3}+\frac {1}{144} x^{4}+\frac {1}{2880} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (1-\frac {3}{4} x^{2}-\frac {7}{36} x^{3}-\frac {35}{1728} x^{4}-\frac {101}{86400} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.018 (sec). Leaf size: 85

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

\[ y(x)\to c_1 \left (\frac {1}{144} x \left (x^3+12 x^2+72 x+144\right ) \log (x)+\frac {-47 x^4-480 x^3-2160 x^2-1728 x+1728}{1728}\right )+c_2 \left (\frac {x^5}{2880}+\frac {x^4}{144}+\frac {x^3}{12}+\frac {x^2}{2}+x\right ) \]