23.5.10 problem 3(i)

Internal problem ID [4187]
Book : Theory and solutions of Ordinary Differential equations, Donald Greenspan, 1960
Section : Chapter 7. Special functions. Exercises at page 124
Problem number : 3(i)
Date solved : Monday, January 27, 2025 at 08:41:32 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-\frac {y^{\prime }}{x}+\left (1+\frac {1}{x^{2}}\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.013 (sec). Leaf size: 34

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

\[ y \left (x \right ) = x \left (\left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1-\frac {1}{4} x^{2}+\frac {1}{64} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\left (\frac {1}{4} x^{2}-\frac {3}{128} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \right ) \]

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 66

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

\[ \{y(x)\}\to c_1 x \left (\frac {x^4}{64}-\frac {x^2}{4}+1\right )+c_2 \left (x \left (\frac {x^2}{4}-\frac {3 x^4}{128}\right )+x \left (\frac {x^4}{64}-\frac {x^2}{4}+1\right ) \log (x)\right ) \]