61.31.10 problem 191

Internal problem ID [12612]
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 : 191
Date solved : Monday, March 31, 2025 at 05:53:59 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.221 (sec). Leaf size: 169

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

\[ y = \frac {\left (a x +b \right )^{\frac {3 a -c}{a}} \left (c_1 \,x^{\frac {-3 a +c}{a}} \operatorname {HeunC}\left (\frac {a \lambda }{b}, 1-\frac {c}{a}, 3-\frac {c}{a}, 0, -\frac {a \lambda }{b}+\frac {c \lambda }{2 b}+\frac {5}{2}-\frac {2 c}{a}+\frac {c^{2}}{2 a^{2}}, -\frac {b}{a x}\right ) x^{2}+c_2 \operatorname {HeunC}\left (\frac {a \lambda }{b}, -1+\frac {c}{a}, 3-\frac {c}{a}, 0, -\frac {a \lambda }{b}+\frac {c \lambda }{2 b}+\frac {5}{2}-\frac {2 c}{a}+\frac {c^{2}}{2 a^{2}}, -\frac {b}{a x}\right )\right )}{x^{2}} \]

✓ Mathematica. Time used: 0.739 (sec). Leaf size: 70

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

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

✗ Sympy

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

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