56.4.60 problem 57

Internal problem ID [8949]
Book : Own collection of miscellaneous problems
Section : section 4.0
Problem number : 57
Date solved : Monday, January 27, 2025 at 05:20:39 PM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} x^{2} y^{\prime \prime }+3 x y^{\prime }+4 x^{4} y&=0 \end{align*}

Using series method with expansion around

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

Solution by Maple

Time used: 0.016 (sec). Leaf size: 28

Order:=6; 
dsolve(x^2*diff(y(x),x$2)+3*x*diff(y(x),x)+4*x^4*y(x) = 0,y(x),type='series',x=0);
 
\[ y = c_{1} \left (1-\frac {1}{6} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_{2} \left (-2+x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{x^{2}} \]

Solution by Mathematica

Time used: 0.010 (sec). Leaf size: 30

AsymptoticDSolveValue[x^2*D[y[x],{x,2}]+3*x*D[y[x],x]+4*x^4*y[x]==0,y[x],{x,0,"6"-1}]
 
\[ y(x)\to c_2 \left (1-\frac {x^4}{6}\right )+c_1 \left (\frac {1}{x^2}-\frac {x^2}{2}\right ) \]