59.1.67 problem 69

Internal problem ID [9239]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 69
Date solved : Wednesday, March 05, 2025 at 07:45:40 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+x^{5} y^{\prime }+6 x^{4} y&=0 \end{align*}

✓ Maple. Time used: 0.007 (sec). Leaf size: 63

ode:=diff(diff(y(x),x),x)+x^5*diff(y(x),x)+6*x^4*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = c_{2} {\mathrm e}^{-\frac {x^{6}}{6}} \left (-x^{6}\right )^{{1}/{6}} 6^{{5}/{6}} \Gamma \left (\frac {5}{6}\right )-c_{2} {\mathrm e}^{-\frac {x^{6}}{6}} \left (-x^{6}\right )^{{1}/{6}} 6^{{5}/{6}} \Gamma \left (\frac {5}{6}, -\frac {x^{6}}{6}\right )+c_{1} {\mathrm e}^{-\frac {x^{6}}{6}} x +6 c_{2} \]

✓ Mathematica. Time used: 0.233 (sec). Leaf size: 53

ode=D[y[x],{x,2}]+x^5*D[y[x],x]+6*x^4*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {1}{36} e^{-\frac {x^6}{6}} \left (36 c_1 x-6^{5/6} c_2 \sqrt [6]{-x^6} \Gamma \left (-\frac {1}{6},-\frac {x^6}{6}\right )\right ) \]

✓ Sympy. Time used: 0.822 (sec). Leaf size: 3

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**5*Derivative(y(x), x) + 6*x**4*y(x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = O\left (1\right ) \]