61.28.27 problem 87

Internal problem ID [12508]
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 : 87
Date solved : Monday, March 31, 2025 at 05:37:06 AM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

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

✓ Maple. Time used: 0.022 (sec). Leaf size: 42

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

\[ y = x \,{\mathrm e}^{-\frac {x \left (a \,x^{2}+3 b \right )}{3}} \left (c_1 \int \frac {{\mathrm e}^{\frac {x \left (a \,x^{2}+3 b \right )}{3}}}{x^{2}}d x +c_2 \right ) \]

✓ Mathematica. Time used: 0.511 (sec). Leaf size: 93

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

\[ y(x)\to \exp \left (\int _1^x\left (-\frac {1}{2} a K[1]^2-\frac {b}{2}+\frac {1}{K[1]}\right )dK[1]-\frac {1}{6} x \left (a x^2+3 b\right )\right ) \left (c_2 \int _1^x\exp \left (-2 \int _1^{K[2]}\left (-\frac {1}{2} a K[1]^2-\frac {b}{2}+\frac {1}{K[1]}\right )dK[1]\right )dK[2]+c_1\right ) \]

✗ Sympy

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

False