43.2.3 problem 7.3.5

Internal problem ID [6860]
Book : Notes on Diffy Qs. Differential Equations for Engineers. By by Jiri Lebl, 2013.
Section : Chapter 7. POWER SERIES METHODS. 7.3.2 The method of Frobenius. Exercises. page 300
Problem number : 7.3.5
Date solved : Monday, January 27, 2025 at 02:31:50 PM
CAS classification : [[_Emden, _Fowler]]

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

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 28

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

\[ y = \left (c_2 \ln \left (x \right )+c_1 \right ) \left (1+\frac {1}{9} x^{3}+\operatorname {O}\left (x^{6}\right )\right )+\left (-\frac {2}{27} x^{3}+\operatorname {O}\left (x^{6}\right )\right ) c_2 \]

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 39

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

\[ y(x)\to c_1 \left (\frac {x^3}{9}+1\right )+c_2 \left (\left (\frac {x^3}{9}+1\right ) \log (x)-\frac {2 x^3}{27}\right ) \]