23.3.316 problem 318

Internal problem ID [6030]
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 : 318
Date solved : Friday, October 03, 2025 at 01:45:52 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.057 (sec). Leaf size: 114

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

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

✓ Mathematica. Time used: 0.165 (sec). Leaf size: 243

ode=c*y[x] + (a + b*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 -i^{-\sqrt {b^2-2 b-4 c+1}+b+1} a^{\frac {1}{2} \left (-\sqrt {b^2-2 b-4 c+1}+b-1\right )} \left (\frac {1}{x}\right )^{\frac {1}{2} \left (-\sqrt {b^2-2 b-4 c+1}+b-1\right )} \left (c_2 i^{2 \sqrt {b^2-2 b-4 c+1}} a^{\sqrt {b^2-2 b-4 c+1}} \left (\frac {1}{x}\right )^{\sqrt {b^2-2 b-4 c+1}} \operatorname {Hypergeometric1F1}\left (\frac {1}{2} \left (b+\sqrt {b^2-2 b-4 c+1}-1\right ),\sqrt {b^2-2 b-4 c+1}+1,\frac {a}{x}\right )+c_1 \operatorname {Hypergeometric1F1}\left (\frac {1}{2} \left (b-\sqrt {b^2-2 b-4 c+1}-1\right ),1-\sqrt {b^2-2 b-4 c+1},\frac {a}{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(c*y(x) + x**2*Derivative(y(x), (x, 2)) + (a + b*x)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - (-c*y(x) - x**2*Derivative(y(x), (x, 2)))/