23.3.405 problem 409

Internal problem ID [6119]
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 : 409
Date solved : Friday, October 03, 2025 at 01:46:31 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.016 (sec). Leaf size: 124

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

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

✓ Mathematica. Time used: 0.118 (sec). Leaf size: 131

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

\begin{align*} y(x)&\to 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 )+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 ) \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*(x + 1)*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