61.28.28 problem 88

Internal problem ID [12509]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-3
Problem number : 88
Date solved : Monday, March 31, 2025 at 05:37:08 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x y^{\prime \prime }+\left (a \,x^{3}+b \,x^{2}+2\right ) y^{\prime }+b x y&=0 \end{align*}

✓ Maple. Time used: 0.097 (sec). Leaf size: 138

ode:=x*diff(diff(y(x),x),x)+(a*x^3+b*x^2+2)*diff(y(x),x)+b*x*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {{\mathrm e}^{-\frac {x^{2} \left (a x +\frac {3 b}{2}\right ) \left (\operatorname {csgn}\left (a \right )+1\right )}{6}} \left (c_2 \,{\mathrm e}^{\frac {\operatorname {csgn}\left (a \right ) x^{2} \left (2 a x +3 b \right )}{6}} \operatorname {HeunT}\left (\frac {3^{{2}/{3}} b}{2 \left (a^{2}\right )^{{1}/{3}}}, 6 \,\operatorname {csgn}\left (a \right ), -\frac {b^{2} 3^{{1}/{3}}}{4 \left (a^{2}\right )^{{2}/{3}}}, -\frac {3^{{2}/{3}} a \left (2 a x +b \right )}{6 \left (a^{2}\right )^{{5}/{6}}}\right )+c_1 \operatorname {HeunT}\left (\frac {3^{{2}/{3}} b}{2 \left (a^{2}\right )^{{1}/{3}}}, -6 \,\operatorname {csgn}\left (a \right ), -\frac {b^{2} 3^{{1}/{3}}}{4 \left (a^{2}\right )^{{2}/{3}}}, \frac {3^{{2}/{3}} a \left (2 a x +b \right )}{6 \left (a^{2}\right )^{{5}/{6}}}\right )\right )}{x} \]

✓ Mathematica. Time used: 0.439 (sec). Leaf size: 76

ode=x*D[y[x],{x,2}]+(a*x^3+b*x^2+2)*D[y[x],x]+b*x*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {(a x+b) \left (c_2 \int _1^x\exp \left (\int _1^{K[2]}-\frac {a^2 K[1]^3+2 a b K[1]^2+b^2 K[1]+2 a}{b+a K[1]}dK[1]\right )dK[2]+c_1\right )}{b x} \]

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
y = Function("y") 
ode = Eq(b*x*y(x) + x*Derivative(y(x), (x, 2)) + (a*x**3 + b*x**2 + 2)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False