78.20.7 problem 5

Internal problem ID [18444]
Book : DIFFERENTIAL EQUATIONS WITH APPLICATIONS AND HISTORICAL NOTES by George F. Simmons. 3rd edition. 2017. CRC press, Boca Raton FL.
Section : Chapter 5. Power Series Solutions and Special Functions. Section 30. Regular singular Points (continued). Problems at page 235
Problem number : 5
Date solved : Tuesday, January 28, 2025 at 11:50:09 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-\frac {1}{4}\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 34

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

\[ y = \frac {c_{1} \left (1-\frac {1}{6} x^{2}+\frac {1}{120} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) x +c_{2} \left (1-\frac {1}{2} x^{2}+\frac {1}{24} x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{\sqrt {x}} \]

✓ Solution by Mathematica

Time used: 0.010 (sec). Leaf size: 58

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

\[ y(x)\to c_1 \left (\frac {x^{7/2}}{24}-\frac {x^{3/2}}{2}+\frac {1}{\sqrt {x}}\right )+c_2 \left (\frac {x^{9/2}}{120}-\frac {x^{5/2}}{6}+\sqrt {x}\right ) \]