42.1.15 problem 3.25 v=1/2

Internal problem ID [6837]
Book : Advanced Mathematical Methods for Scientists and Engineers, Bender and Orszag. Springer October 29, 1999
Section : Chapter 3. APPROXIMATE SOLUTION OF LINEAR DIFFERENTIAL EQUATIONS. page 136
Problem number : 3.25 v=1/2
Date solved : Monday, January 27, 2025 at 02:31:27 PM
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: 0.006 (sec). Leaf size: 35

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

\[ y = c_1 \,x^{-\frac {i}{2}} \left (1+\left (-\frac {1}{5}-\frac {i}{10}\right ) x^{2}+\left (\frac {7}{680}+\frac {3 i}{340}\right ) x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \,x^{\frac {i}{2}} \left (1+\left (-\frac {1}{5}+\frac {i}{10}\right ) x^{2}+\left (\frac {7}{680}-\frac {3 i}{340}\right ) x^{4}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.009 (sec). Leaf size: 66

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}]
 

\[ y(x)\to \left (\frac {7}{680}-\frac {3 i}{340}\right ) c_1 x^{1 i/2} \left (x^4-(16+4 i) x^2+(56+48 i)\right )+\left (\frac {7}{680}+\frac {3 i}{340}\right ) c_2 x^{-i/2} \left (x^4-(16-4 i) x^2+(56-48 i)\right ) \]