23.3.498 problem 504

Internal problem ID [6212]
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 : 504
Date solved : Friday, October 03, 2025 at 01:56:56 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.035 (sec). Leaf size: 143

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

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

✓ Mathematica. Time used: 0.216 (sec). Leaf size: 145

ode=c*x*y[x] + (a + b*x^2)*D[y[x],x] + x*(1 + x^2)*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 (b-\sqrt {b^2-2 b-4 c+1}-1\right ),\frac {1}{4} \left (b+\sqrt {b^2-2 b-4 c+1}-1\right ),\frac {a+1}{2},-x^2\right )+c_2 x^{1-a} \operatorname {Hypergeometric2F1}\left (\frac {1}{4} \left (-2 a+b-\sqrt {b^2-2 b-4 c+1}+1\right ),\frac {1}{4} \left (-2 a+b+\sqrt {b^2-2 b-4 c+1}+1\right ),\frac {3-a}{2},-x^2\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*x*y(x) + x*(x**2 + 1)*Derivative(y(x), (x, 2)) + (a + b*x**2)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

ValueError : Expected Expr or iterable but got None