42.1.16 problem 3.25 v=3/2

Internal problem ID [6838]
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=3/2
Date solved : Monday, January 27, 2025 at 02:31:28 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}+\frac {9}{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+(3/2)^2)*y(x)=0,y(x),type='series',x=0);
 

\[ y = c_1 \,x^{-\frac {3 i}{2}} \left (1+\left (-\frac {1}{13}-\frac {3 i}{26}\right ) x^{2}+\left (-\frac {1}{2600}+\frac {9 i}{1300}\right ) x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \,x^{\frac {3 i}{2}} \left (1+\left (-\frac {1}{13}+\frac {3 i}{26}\right ) x^{2}+\left (-\frac {1}{2600}-\frac {9 i}{1300}\right ) x^{4}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ 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+9/4)*y[x]==0,y[x],{x,0,"6"-1}]
 

\[ y(x)\to \left (-\frac {1}{2600}-\frac {9 i}{1300}\right ) c_1 x^{3 i/2} \left (x^4-(16+12 i) x^2-(8-144 i)\right )-\left (\frac {1}{2600}-\frac {9 i}{1300}\right ) c_2 x^{-3 i/2} \left (x^4-(16-12 i) x^2-(8+144 i)\right ) \]