23.3.409 problem 413

Internal problem ID [6123]
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 : 413
Date solved : Friday, October 03, 2025 at 01:46:34 AM
CAS classification : [_Jacobi]

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

✓ Maple. Time used: 0.016 (sec). Leaf size: 116

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

\[ y = c_1 \operatorname {hypergeom}\left (\left [-\frac {1}{2}-\frac {b}{2}-\frac {\sqrt {b^{2}+2 b +4 c +1}}{2}, -\frac {1}{2}-\frac {b}{2}+\frac {\sqrt {b^{2}+2 b +4 c +1}}{2}\right ], \left [a \right ], x\right )+c_2 \,x^{-a +1} \operatorname {hypergeom}\left (\left [\frac {1}{2}-\frac {b}{2}-\frac {\sqrt {b^{2}+2 b +4 c +1}}{2}-a , \frac {1}{2}-\frac {b}{2}+\frac {\sqrt {b^{2}+2 b +4 c +1}}{2}-a \right ], \left [2-a \right ], x\right ) \]

✓ Mathematica. Time used: 0.11 (sec). Leaf size: 141

ode=c*y[x] + (a + b*x)*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_1 \operatorname {Hypergeometric2F1}\left (\frac {1}{2} \left (-b-\sqrt {b^2+2 b+4 c+1}-1\right ),\frac {1}{2} \left (-b+\sqrt {b^2+2 b+4 c+1}-1\right ),a,x\right )-(-1)^{-a} c_2 x^{1-a} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} \left (-2 a-b-\sqrt {b^2+2 b+4 c+1}+1\right ),\frac {1}{2} \left (-2 a-b+\sqrt {b^2+2 b+4 c+1}+1\right ),2-a,x\right ) \end{align*}

✗ Sympy

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

ValueError : Expected Expr or iterable but got None