50.20.5 problem 3(c)

Internal problem ID [8133]
Book : Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section : Chapter 4. Power Series Solutions and Special Functions. Section 4.5. More on Regular Singular Points. Page 183
Problem number : 3(c)
Date solved : Monday, January 27, 2025 at 03:44:27 PM
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: 0.018 (sec). Leaf size: 28

Order:=8; 
dsolve(x*diff(y(x),x$2)-diff(y(x),x)+4*x^3*y(x)=0,y(x),type='series',x=0);
 

\[ y = c_{1} x^{2} \left (1-\frac {1}{6} x^{4}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} \left (-2+x^{4}+\operatorname {O}\left (x^{8}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.010 (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,"8"-1}]
 

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