61.31.2 problem 183

Internal problem ID [12604]
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 : 183
Date solved : Thursday, March 13, 2025 at 11:52:59 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.463 (sec). Leaf size: 120

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

\[ y = x^{-\frac {\sqrt {a^{2}-2 a -4 c +1}}{2}-\frac {a}{2}+\frac {1}{2}} \left (\operatorname {KummerU}\left (-\frac {1}{4}+\frac {\sqrt {a^{2}-2 a -4 c +1}}{4}+\frac {a}{4}, 1+\frac {\sqrt {a^{2}-2 a -4 c +1}}{2}, \frac {b}{2 x^{2}}\right ) c_{2} +\operatorname {KummerM}\left (-\frac {1}{4}+\frac {\sqrt {a^{2}-2 a -4 c +1}}{4}+\frac {a}{4}, 1+\frac {\sqrt {a^{2}-2 a -4 c +1}}{2}, \frac {b}{2 x^{2}}\right ) c_{1} \right ) \]

✓ Mathematica. Time used: 0.389 (sec). Leaf size: 308

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

\[ y(x)\to -(-1)^{\frac {1}{4} \left (-\sqrt {a^2-2 a-4 c+1}+a+3\right )} 2^{\frac {1}{4} \left (-\sqrt {a^2-2 a-4 c+1}-a+1\right )} b^{\frac {1}{4} \left (-\sqrt {a^2-2 a-4 c+1}+a-1\right )} \left (\frac {1}{x}\right )^{\frac {1}{2} \left (-\sqrt {a^2-2 a-4 c+1}+a-1\right )} \left (c_2 i^{\sqrt {a^2-2 a-4 c+1}} b^{\frac {1}{2} \sqrt {a^2-2 a-4 c+1}} \left (\frac {1}{x}\right )^{\sqrt {a^2-2 a-4 c+1}} \operatorname {Hypergeometric1F1}\left (\frac {1}{4} \left (a+\sqrt {a^2-2 a-4 c+1}-1\right ),\frac {1}{2} \left (\sqrt {a^2-2 a-4 c+1}+2\right ),\frac {b}{2 x^2}\right )+c_1 2^{\frac {1}{2} \sqrt {a^2-2 a-4 c+1}} \operatorname {Hypergeometric1F1}\left (\frac {1}{4} \left (a-\sqrt {a^2-2 a-4 c+1}-1\right ),1-\frac {1}{2} \sqrt {a^2-2 a-4 c+1},\frac {b}{2 x^2}\right )\right ) \]

✗ Sympy

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

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