23.3.291 problem 293

Internal problem ID [6005]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 3. THE DIFFERENTIAL EQUATION IS LINEAR AND OF SECOND ORDER, page 311
Problem number : 293
Date solved : Friday, October 03, 2025 at 01:45:49 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.010 (sec). Leaf size: 47

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

\[ y = {\mathrm e}^{-\frac {1}{x}} \sqrt {\frac {1}{x}}\, \left (\operatorname {BesselK}\left (\frac {\sqrt {1-4 a}}{2}, \frac {1}{x}\right ) c_2 +\operatorname {BesselI}\left (\frac {\sqrt {1-4 a}}{2}, \frac {1}{x}\right ) c_1 \right ) \]

✓ Mathematica. Time used: 0.08 (sec). Leaf size: 145

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

\begin{align*} y(x)&\to 2^{\frac {1}{2}-\frac {1}{2} \sqrt {1-4 a}} \left (\frac {1}{x}\right )^{\frac {1}{2}-\frac {1}{2} \sqrt {1-4 a}} \left (2^{\sqrt {1-4 a}} c_2 \left (\frac {1}{x}\right )^{\sqrt {1-4 a}} \operatorname {Hypergeometric1F1}\left (\frac {1}{2} \left (\sqrt {1-4 a}+1\right ),\sqrt {1-4 a}+1,-\frac {2}{x}\right )+c_1 \operatorname {Hypergeometric1F1}\left (\frac {1}{2}-\frac {1}{2} \sqrt {1-4 a},1-\sqrt {1-4 a},-\frac {2}{x}\right )\right ) \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE Derivative(y(x), x) - (-a*y(x) - x**2*Derivative(y(x), (x, 2)))/