56.4.61 problem 58

Internal problem ID [8950]
Book : Own collection of miscellaneous problems
Section : section 4.0
Problem number : 58
Date solved : Monday, January 27, 2025 at 05:20:40 PM
CAS classification : [[_Emden, _Fowler]]

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

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.014 (sec). Leaf size: 58

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

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

✓ Solution by Mathematica

Time used: 0.019 (sec). Leaf size: 85

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

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