43.2.5 problem 7.3.7

Internal problem ID [6862]
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 : 7.3.7
Date solved : Monday, January 27, 2025 at 02:31:53 PM
CAS classification : [[_Emden, _Fowler]]

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

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 28

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

\[ y = x \left (x^{-\sqrt {2}} c_1 +x^{\sqrt {2}} c_2 \right )+O\left (x^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 30

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

\[ y(x)\to c_1 x^{1+\sqrt {2}}+c_2 x^{1-\sqrt {2}} \]