23.5.9 problem 3(h)

Internal problem ID [4186]
Book : Theory and solutions of Ordinary Differential equations, Donald Greenspan, 1960
Section : Chapter 7. Special functions. Exercises at page 124
Problem number : 3(h)
Date solved : Monday, January 27, 2025 at 08:41:31 AM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} y^{\prime \prime }-\frac {y^{\prime }}{2 x}+\frac {y}{4 x}&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.013 (sec). Leaf size: 44

Order:=6; 
dsolve(diff(y(x),x$2)-1/(2*x)*diff(y(x),x)+1/(4*x)*y(x) =0,y(x),type='series',x=0);
 

\[ y \left (x \right ) = c_{1} x^{{3}/{2}} \left (1-\frac {1}{10} x +\frac {1}{280} x^{2}-\frac {1}{15120} x^{3}+\frac {1}{1330560} x^{4}-\frac {1}{172972800} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (1+\frac {1}{2} x -\frac {1}{8} x^{2}+\frac {1}{144} x^{3}-\frac {1}{5760} x^{4}+\frac {1}{403200} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 86

AsymptoticDSolveValue[D[y[x],{x,2}]-1/(2*x)*D[y[x],x]+1/(4*x)*y[x] ==0,{y[x]},{x,0,"6"-1}]
 

\[ \{y(x)\}\to c_2 \left (\frac {x^5}{403200}-\frac {x^4}{5760}+\frac {x^3}{144}-\frac {x^2}{8}+\frac {x}{2}+1\right )+c_1 \left (-\frac {x^5}{172972800}+\frac {x^4}{1330560}-\frac {x^3}{15120}+\frac {x^2}{280}-\frac {x}{10}+1\right ) x^{3/2} \]