53.1.66 problem 68

Internal problem ID [10538]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 68
Date solved : Tuesday, September 30, 2025 at 07:28:29 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (x^{8}+1\right ) y^{\prime \prime }-16 x^{7} y^{\prime }+72 x^{6} y&=0 \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 22

ode:=(x^8+1)*diff(diff(y(x),x),x)-16*x^7*diff(y(x),x)+72*x^6*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = -\frac {7}{9} c_1 +c_1 \,x^{8}+c_2 \,x^{9}-\frac {9}{7} c_2 x \]

✓ Mathematica. Time used: 51.192 (sec). Leaf size: 342

ode=(1+x^8)*D[y[x],{x,2}]-16*x^7*D[y[x],x]+72*x^6*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -\frac {1}{7} \left (-7 x^6-12 (-1)^{3/8} x^5-15 (-1)^{3/4} x^4+16 \sqrt [8]{-1} x^3+15 i x^2+12 (-1)^{7/8} x-7 \sqrt [4]{-1}\right ) \left (x^8+1\right ) \exp \left (\int _1^x\frac {K[1] \left (3 \sqrt [4]{-1}-K[1] \left (K[1] \left (5 K[1]^4-3 (-1)^{3/8} K[1]^3-3 (-1)^{3/4} K[1]^2+3 \sqrt [8]{-1} K[1]+3 i\right )+3 (-1)^{7/8}\right )\right )+3 (-1)^{5/8}}{K[1]^8+1}dK[1]\right ) \left (c_2 \int _1^x\frac {49 \exp \left (-2 \int _1^{K[2]}\frac {K[1] \left (3 \sqrt [4]{-1}-K[1] \left (K[1] \left (5 K[1]^4-3 (-1)^{3/8} K[1]^3-3 (-1)^{3/4} K[1]^2+3 \sqrt [8]{-1} K[1]+3 i\right )+3 (-1)^{7/8}\right )\right )+3 (-1)^{5/8}}{K[1]^8+1}dK[1]\right )}{\left (7 K[2]^6+12 (-1)^{3/8} K[2]^5+15 (-1)^{3/4} K[2]^4-16 \sqrt [8]{-1} K[2]^3-15 i K[2]^2-12 (-1)^{7/8} K[2]+7 \sqrt [4]{-1}\right )^2}dK[2]+c_1\right ) \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-16*x**7*Derivative(y(x), x) + 72*x**6*y(x) + (x**8 + 1)*Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False