61.31.16 problem 197

Internal problem ID [12618]
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 : 197
Date solved : Thursday, March 13, 2025 at 11:53:24 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (a \,x^{3}+b \,x^{2}+c x \right ) y^{\prime \prime }+\left (-2 a \,x^{2}-\left (b +1\right ) x +k \right ) y^{\prime }+2 \left (a x +1\right ) y&=0 \end{align*}

✓ Maple. Time used: 0.983 (sec). Leaf size: 2265

ode:=(a*x^3+b*x^2+c*x)*diff(diff(y(x),x),x)+(-2*a*x^2-(b+1)*x+k)*diff(y(x),x)+2*(a*x+1)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ \text {Expression too large to display} \]

✓ Mathematica. Time used: 5.846 (sec). Leaf size: 291

ode=(a*x^3+b*x^2+c*x)*D[y[x],{x,2}]+(-2*a*x^2-(b+1)*x+k)*D[y[x],x]+2*(a*x+1)*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to -\frac {\left (-k x (a x+2)-(b-1) x^2+c (k-2 x)+k^2\right ) \left (c_2 \int _1^x\exp \left (\int _1^{K[2]}\frac {k^3-K[1] (b+3 a K[1]+3) k^2-K[1]^2 (a K[1]+1) (3 b+2 a K[1]-3) k-4 c^2 K[1]-(b-1) K[1]^3 (3 b+2 a K[1]-1)+c \left (k^2-K[1] (b+6 a K[1]+7) k-6 (b-1) K[1]^2\right )}{K[1] \left (-\left ((k-2 K[1]) c^2\right )-\left (k^2+(b-2) K[1] k+K[1]^2 (-3 b-2 a K[1]+1)\right ) c+K[1] \left (b^2 K[1]^2+a \left (-k^2+K[1] (a K[1]+2) k-K[1]^2\right ) K[1]+b \left (-k^2+K[1] (a K[1]+2) k+K[1]^2 (a K[1]-1)\right )\right )\right )}dK[1]\right )dK[2]+c_1\right )}{a k+b-c (k-2)-k^2+2 k-1} \]

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
c = symbols("c") 
k = symbols("k") 
y = Function("y") 
ode = Eq((2*a*x + 2)*y(x) + (-2*a*x**2 + k - x*(b + 1))*Derivative(y(x), x) + (a*x**3 + b*x**2 + c*x)*Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

ValueError : Expected Expr or iterable but got None