61.31.5 problem 186

Internal problem ID [12607]
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 : 186
Date solved : Thursday, March 13, 2025 at 11:53:03 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.317 (sec). Leaf size: 132

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

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

✓ Mathematica. Time used: 0.359 (sec). Leaf size: 255

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

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

✗ Sympy

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

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