23.3.482 problem 488

Internal problem ID [6196]
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 : 488
Date solved : Friday, October 03, 2025 at 01:56:51 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (\operatorname {b2} x +\operatorname {a2} \right ) y+\operatorname {a1} x y^{\prime }+x^{3} y^{\prime \prime }&=0 \end{align*}

✓ Maple. Time used: 0.038 (sec). Leaf size: 95

ode:=(b2*x+a2)*y(x)+a1*x*diff(y(x),x)+x^3*diff(diff(y(x),x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = x^{\frac {1}{2}-\frac {\sqrt {1-4 \operatorname {b2}}}{2}} \left (\operatorname {KummerU}\left (\frac {\sqrt {1-4 \operatorname {b2}}\, \operatorname {a1} -\operatorname {a1} -2 \operatorname {a2}}{2 \operatorname {a1}}, 1+\sqrt {1-4 \operatorname {b2}}, \frac {\operatorname {a1}}{x}\right ) c_2 +\operatorname {KummerM}\left (\frac {\sqrt {1-4 \operatorname {b2}}\, \operatorname {a1} -\operatorname {a1} -2 \operatorname {a2}}{2 \operatorname {a1}}, 1+\sqrt {1-4 \operatorname {b2}}, \frac {\operatorname {a1}}{x}\right ) c_1 \right ) \]

✓ Mathematica. Time used: 0.166 (sec). Leaf size: 190

ode=(a2 + b2*x)*y[x] + a1*x*D[y[x],x] + x^3*D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -i i^{-\sqrt {1-4 \text {b2}}} \text {a1}^{-\frac {1}{2} \sqrt {1-4 \text {b2}}-\frac {1}{2}} \left (\frac {1}{x}\right )^{-\frac {1}{2} \sqrt {1-4 \text {b2}}-\frac {1}{2}} \left (i^{2 \sqrt {1-4 \text {b2}}} c_2 \text {a1}^{\sqrt {1-4 \text {b2}}} \left (\frac {1}{x}\right )^{\sqrt {1-4 \text {b2}}} \operatorname {Hypergeometric1F1}\left (\frac {1}{2} \left (-\frac {2 \text {a2}}{\text {a1}}+\sqrt {1-4 \text {b2}}-1\right ),\sqrt {1-4 \text {b2}}+1,\frac {\text {a1}}{x}\right )+c_1 \operatorname {Hypergeometric1F1}\left (-\frac {\sqrt {1-4 \text {b2}} \text {a1}+\text {a1}+2 \text {a2}}{2 \text {a1}},1-\sqrt {1-4 \text {b2}},\frac {\text {a1}}{x}\right )\right ) \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
a1 = symbols("a1") 
a2 = symbols("a2") 
b2 = symbols("b2") 
y = Function("y") 
ode = Eq(a1*x*Derivative(y(x), x) + x**3*Derivative(y(x), (x, 2)) + (a2 + b2*x)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - (-a2*y(x) + x*(-b2*y(x) - x**2*Derivative(