61.29.28 problem 137

Internal problem ID [12558]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-4
Problem number : 137
Date solved : Monday, March 31, 2025 at 05:38:59 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 51

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

\[ y = -c_2 \,{\mathrm e}^{-a x} \left (\Gamma \left (b , -a x \right ) b -\Gamma \left (1+b \right )\right ) \left (-a x \right )^{-b}+c_1 \,x^{-b} {\mathrm e}^{-a x}-c_2 \]

✓ Mathematica. Time used: 0.096 (sec). Leaf size: 48

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

\[ y(x)\to e^{-a x} \left (\frac {c_1 (-a x)^{-b} \Gamma (b+1,-a x)}{a}+e^{-b} c_2 x^{-b}\right ) \]

✓ Sympy. Time used: 2.743 (sec). Leaf size: 1110

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

\[ \text {Solution too large to show} \]