56.4.60 problem 57

Internal problem ID [8949]
Book : Own collection of miscellaneous problems
Section : section 4.0
Problem number : 57
Date solved : Sunday, March 30, 2025 at 01:56:10 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*}

✓ Maple. Time used: 0.027 (sec). Leaf size: 28

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

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

✓ Mathematica. Time used: 0.01 (sec). Leaf size: 30

ode=x^2*D[y[x],{x,2}]+3*x*D[y[x],x]+4*x^4*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

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

✓ Sympy. Time used: 0.944 (sec). Leaf size: 26

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(4*x**4*y(x) + x**2*Derivative(y(x), (x, 2)) + 3*x*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

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