23.3.276 problem 278

Internal problem ID [5990]
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 : 278
Date solved : Friday, October 03, 2025 at 01:45:44 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.042 (sec). Leaf size: 45

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

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

✓ Mathematica. Time used: 0.032 (sec). Leaf size: 104

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

\begin{align*} y(x)&\to x^{\sqrt {a}} e^{-\sqrt {c} x} \left (c_1 \operatorname {HypergeometricU}\left (\frac {b}{2 \sqrt {c}}+\sqrt {a}+\frac {1}{2},2 \sqrt {a}+1,2 \sqrt {c} x\right )+c_2 L_{-\frac {b}{2 \sqrt {c}}-\sqrt {a}-\frac {1}{2}}^{2 \sqrt {a}}\left (2 \sqrt {c} x\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(x**2*Derivative(y(x), (x, 2)) + x*Derivative(y(x), x) + (-a - b*x - c*x**2)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

ValueError : Expected Expr or iterable but got None