23.3.438 problem 443

Internal problem ID [6152]
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 : 443
Date solved : Friday, October 03, 2025 at 01:48:02 AM
CAS classification : [_Jacobi]

\begin{align*} 2 a \left (1+a \right ) y-\left (1+3 x \right ) y^{\prime }+2 \left (1-x \right ) x y^{\prime \prime }&=0 \end{align*}

✓ Maple. Time used: 0.020 (sec). Leaf size: 86

ode:=2*a*(a+1)*y(x)-(3*x+1)*diff(y(x),x)+2*(1-x)*x*diff(diff(y(x),x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = c_1 \operatorname {hypergeom}\left (\left [\frac {1}{4}-\frac {\sqrt {16 a^{2}+16 a +1}}{4}, \frac {1}{4}+\frac {\sqrt {16 a^{2}+16 a +1}}{4}\right ], \left [-\frac {1}{2}\right ], x\right )+c_2 \,x^{{3}/{2}} \operatorname {hypergeom}\left (\left [\frac {7}{4}+\frac {\sqrt {16 a^{2}+16 a +1}}{4}, \frac {7}{4}-\frac {\sqrt {16 a^{2}+16 a +1}}{4}\right ], \left [\frac {5}{2}\right ], x\right ) \]

✓ Mathematica. Time used: 0.076 (sec). Leaf size: 114

ode=2*a*(1 + a)*y[x] - (1 + 3*x)*D[y[x],x] + 2*(1 - x)*x*D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to c_1 \operatorname {Hypergeometric2F1}\left (\frac {1}{4} \left (1-\sqrt {16 a^2+16 a+1}\right ),\frac {1}{4} \left (\sqrt {16 a^2+16 a+1}+1\right ),-\frac {1}{2},x\right )-i c_2 x^{3/2} \operatorname {Hypergeometric2F1}\left (\frac {1}{4} \left (7-\sqrt {16 a^2+16 a+1}\right ),\frac {1}{4} \left (\sqrt {16 a^2+16 a+1}+7\right ),\frac {5}{2},x\right ) \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(2*a*(a + 1)*y(x) + x*(2 - 2*x)*Derivative(y(x), (x, 2)) - (3*x + 1)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False