59.1.84 problem 86

Internal problem ID [9256]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 86
Date solved : Sunday, March 30, 2025 at 02:30:48 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 12 x^{2} \left (1+x \right ) y^{\prime \prime }+x \left (3 x^{2}+35 x +11\right ) y^{\prime }-\left (-5 x^{2}-10 x +1\right ) y&=0 \end{align*}

✓ Maple. Time used: 0.155 (sec). Leaf size: 43

ode:=12*x^2*(1+x)*diff(diff(y(x),x),x)+x*(3*x^2+35*x+11)*diff(y(x),x)-(-5*x^2-10*x+1)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {{\mathrm e}^{-\frac {x}{4}} \left (x^{{7}/{12}} \operatorname {HeunC}\left (\frac {1}{4}, \frac {7}{12}, -\frac {3}{4}, -\frac {1}{12}, \frac {1}{2}, -x \right ) c_2 +\operatorname {HeunC}\left (\frac {1}{4}, -\frac {7}{12}, -\frac {3}{4}, -\frac {1}{12}, \frac {1}{2}, -x \right ) c_1 \right )}{\left (1+x \right )^{{3}/{4}} x^{{1}/{4}}} \]

✓ Mathematica. Time used: 0.37 (sec). Leaf size: 118

ode=12*x^2*(1+x)*D[y[x],{x,2}]+x*(11+35*x+3*x^2)*D[y[x],x]-(1-10*x-5*x^2)*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \exp \left (\int _1^x\frac {1}{24} \left (\frac {3}{K[1]+1}-3+\frac {5}{K[1]}\right )dK[1]-\frac {1}{2} \int _1^x\frac {1}{12} \left (\frac {21}{K[2]+1}+3+\frac {11}{K[2]}\right )dK[2]\right ) \left (c_2 \int _1^x\exp \left (-2 \int _1^{K[3]}\frac {-3 K[1]^2+5 K[1]+5}{24 K[1]^2+24 K[1]}dK[1]\right )dK[3]+c_1\right ) \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(12*x**2*(x + 1)*Derivative(y(x), (x, 2)) + x*(3*x**2 + 35*x + 11)*Derivative(y(x), x) - (-5*x**2 - 10*x + 1)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False