23.3.383 problem 387

Internal problem ID [6097]
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 : 387
Date solved : Friday, October 03, 2025 at 01:46:24 AM
CAS classification : [_Jacobi]

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

✓ Maple. Time used: 0.024 (sec). Leaf size: 148

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

\[ y = -\frac {3 \left (c_1 \left (x^{2+\frac {\sqrt {5}}{2}}-\frac {x^{3+\frac {\sqrt {5}}{2}}}{3}+\frac {x^{\frac {\sqrt {5}}{2}}}{3}-x^{1+\frac {\sqrt {5}}{2}}\right ) \operatorname {hypergeom}\left (\left [\frac {5}{2}-\frac {\sqrt {5}}{2}, \frac {3}{2}-\frac {\sqrt {5}}{2}\right ], \left [-\sqrt {5}+1\right ], \frac {1}{x}\right )+\operatorname {hypergeom}\left (\left [\frac {5}{2}+\frac {\sqrt {5}}{2}, \frac {3}{2}+\frac {\sqrt {5}}{2}\right ], \left [\sqrt {5}+1\right ], \frac {1}{x}\right ) c_2 \left (x^{2-\frac {\sqrt {5}}{2}}-\frac {x^{3-\frac {\sqrt {5}}{2}}}{3}+\frac {x^{-\frac {\sqrt {5}}{2}}}{3}-x^{1-\frac {\sqrt {5}}{2}}\right )\right )}{x^{{5}/{2}}} \]

✓ Mathematica. Time used: 0.736 (sec). Leaf size: 73

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

\begin{align*} y(x)&\to c_2 G_{2,2}^{2,0}\left (x\left | \begin {array}{c} \frac {1}{2} \left (3-\sqrt {5}\right ),\frac {1}{2} \left (3+\sqrt {5}\right ) \\ -1,0 \\ \end {array} \right .\right )+c_1 \operatorname {Hypergeometric2F1}\left (\frac {1}{2} \left (-1-\sqrt {5}\right ),\frac {1}{2} \left (-1+\sqrt {5}\right ),2,x\right ) \end{align*}

✗ Sympy

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

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