78.20.5 problem 3 (c)

Internal problem ID [18442]
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 : 3 (c)
Date solved : Tuesday, January 28, 2025 at 11:50:07 AM
CAS classification : [[_Emden, _Fowler], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

\begin{align*} x y^{\prime \prime }-y^{\prime }+4 x^{3} 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: 28

Order:=6; 
dsolve(x*diff(y(x),x$2)-diff(y(x),x)+4*x^3*y(x)=0,y(x),type='series',x=0);
 

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

✓ Solution by Mathematica

Time used: 0.007 (sec). Leaf size: 30

AsymptoticDSolveValue[x*D[y[x],{x,2}]-D[y[x],x]+4*x^3*y[x]==0,y[x],{x,0,"6"-1}]
 

\[ y(x)\to c_1 \left (1-\frac {x^4}{2}\right )+c_2 \left (x^2-\frac {x^6}{6}\right ) \]