23.5.5 problem 3(d)

Internal problem ID [4182]
Book : Theory and solutions of Ordinary Differential equations, Donald Greenspan, 1960
Section : Chapter 7. Special functions. Exercises at page 124
Problem number : 3(d)
Date solved : Monday, January 27, 2025 at 08:41:26 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-2 y^{\prime }+\left (\frac {1}{4 x^{2}}-1\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: 46

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

\[ y \left (x \right ) = \left (\left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1+x +x^{2}+\frac {2}{3} x^{3}+\frac {17}{48} x^{4}+\frac {37}{240} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (-\frac {1}{2} x^{2}-\frac {1}{2} x^{3}-\frac {11}{32} x^{4}-\frac {17}{96} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \right ) \sqrt {x} \]

✓ Solution by Mathematica

Time used: 0.005 (sec). Leaf size: 114

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

\[ \{y(x)\}\to c_1 \sqrt {x} \left (\frac {37 x^5}{240}+\frac {17 x^4}{48}+\frac {2 x^3}{3}+x^2+x+1\right )+c_2 \left (\sqrt {x} \left (-\frac {17 x^5}{96}-\frac {11 x^4}{32}-\frac {x^3}{2}-\frac {x^2}{2}\right )+\sqrt {x} \left (\frac {37 x^5}{240}+\frac {17 x^4}{48}+\frac {2 x^3}{3}+x^2+x+1\right ) \log (x)\right ) \]