2.3 problem 7.3.5

Internal problem ID [4764]

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.
ODE order: 2.
ODE degree: 1.

CAS Maple gives this as type [[_Emden, _Fowler]]

Solve \begin {gather*} \boxed {y^{\prime \prime }+\frac {y^{\prime }}{x}-y x=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).

Solution by Maple

Time used: 0.047 (sec). Leaf size: 35

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

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

Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 39

AsymptoticDSolveValue[y''[x]+1/x*y'[x]-x*y[x]==0,y[x],{x,0,5}]
 

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