61.31.6 problem 187

Internal problem ID [12608]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-6
Problem number : 187
Date solved : Thursday, March 13, 2025 at 11:53:05 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 1.634 (sec). Leaf size: 49

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

\[ y = {\mathrm e}^{-a x} \left (c_{2} \left (\int \frac {x \,{\mathrm e}^{\frac {2 a \,x^{3}+2 a x b +b}{2 x^{2}}}}{\left (a x +1\right )^{2}}d x \right )+c_{1} \right ) \left (a x +1\right ) \]

✓ Mathematica. Time used: 0.798 (sec). Leaf size: 138

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

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

✗ Sympy

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

NotImplementedError : The given ODE -x*(-a**2*b*y(x) - x**2*Derivative(y(x), (x, 2)))/(a*b*x + a*x**3 + b - x**2) + Derivative(y(x), x) cannot be solved by the factorable group method