23.3.380 problem 384

Internal problem ID [6094]
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 : 384
Date solved : Friday, October 03, 2025 at 01:46:21 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.020 (sec). Leaf size: 134

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

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

✓ Mathematica. Time used: 0.115 (sec). Leaf size: 184

ode=c*y[x] + (a + b*x)*D[y[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 2^{\frac {1}{2} (-a-b-2)} \left (c_2 (x-1)^{\frac {1}{2} (a+b+2)} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} \left (a-\sqrt {b^2+2 b+4 c+1}+1\right ),\frac {1}{2} \left (a+\sqrt {b^2+2 b+4 c+1}+1\right ),\frac {1}{2} (a+b+4),\frac {1-x}{2}\right )+c_1 2^{\frac {1}{2} (a+b+2)} \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 ),\frac {1}{2} (-a-b),\frac {1-x}{2}\right )\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) + (1 - x**2)*Derivative(y(x), (x, 2)) + (a + b*x)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False