42.1.10 problem 3.24 (e)

Internal problem ID [6832]
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 : 3.24 (e)
Date solved : Monday, January 27, 2025 at 02:31:22 PM
CAS classification : [[_Emden, _Fowler], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

\begin{align*} 2 x y^{\prime \prime }-y^{\prime }+x^{2} y&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 28

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

\[ y = c_1 \,x^{{3}/{2}} \left (1-\frac {1}{27} x^{3}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (1-\frac {1}{9} x^{3}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 33

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

\[ y(x)\to c_2 \left (1-\frac {x^3}{9}\right )+c_1 \left (1-\frac {x^3}{27}\right ) x^{3/2} \]