23.3.234 problem 236

Internal problem ID [5948]
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 : 236
Date solved : Tuesday, September 30, 2025 at 02:06:41 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.047 (sec). Leaf size: 43

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

\[ y = \frac {3 \,2^{{3}/{4}} {\mathrm e}^{\frac {x^{2}}{2}} \left (\left (-c_1 +\frac {4 c_2}{3}\right ) \Gamma \left (\frac {3}{4}, \frac {x^{2}}{2}\right )+c_1 \Gamma \left (\frac {3}{4}\right )\right ) x^{{3}/{2}}}{4 \left (x^{2}\right )^{{3}/{4}}} \]

✓ Mathematica. Time used: 0.145 (sec). Leaf size: 55

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

\begin{align*} y(x)&\to \frac {1}{2} e^{\frac {x^2}{2}} \left (2 c_1-\frac {2^{3/4} c_2 \sqrt [4]{x^2} \Gamma \left (\frac {3}{4},\frac {x^2}{2}\right )}{\sqrt {x}}\right ) \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x*y(x) + 2*x*Derivative(y(x), (x, 2)) - (2*x**2 + 1)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False