59.1.113 problem 115

Internal problem ID [9285]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 115
Date solved : Sunday, March 30, 2025 at 02:31:30 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 8 x^{2} \left (2 x^{2}+1\right ) y^{\prime \prime }+2 x \left (34 x^{2}+5\right ) y^{\prime }-\left (-30 x^{2}+1\right ) y&=0 \end{align*}

✓ Maple. Time used: 0.064 (sec). Leaf size: 46

ode:=8*x^2*(2*x^2+1)*diff(diff(y(x),x),x)+2*x*(34*x^2+5)*diff(y(x),x)-(-30*x^2+1)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {c_1 \operatorname {LegendreP}\left (\frac {3}{8}, \frac {3}{8}, \sqrt {2 x^{2}+1}\right )+c_2 \operatorname {LegendreQ}\left (\frac {3}{8}, \frac {3}{8}, \sqrt {2 x^{2}+1}\right )}{x^{{1}/{8}} \sqrt {2 x^{2}+1}} \]

✓ Mathematica. Time used: 0.359 (sec). Leaf size: 118

ode=8*x^2*(1+2*x^2)*D[y[x],{x,2}]+2*x*(5+34*x^2)*D[y[x],x]-(1-30*x^2)*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \exp \left (\int _1^x\left (\frac {K[1]}{2 K[1]^2+1}+\frac {7}{8 K[1]}\right )dK[1]-\frac {1}{2} \int _1^x\frac {34 K[2]^2+5}{8 K[2]^3+4 K[2]}dK[2]\right ) \left (c_2 \int _1^x\exp \left (-2 \int _1^{K[3]}\left (\frac {K[1]}{2 K[1]^2+1}+\frac {7}{8 K[1]}\right )dK[1]\right )dK[3]+c_1\right ) \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(8*x**2*(2*x**2 + 1)*Derivative(y(x), (x, 2)) + 2*x*(34*x**2 + 5)*Derivative(y(x), x) - (1 - 30*x**2)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False