23.3.489 problem 495
Internal
problem
ID
[6203]
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
:
495
Date
solved
:
Friday, October 03, 2025 at 01:56:55 AM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries]]
\begin{align*} \left (\operatorname {b2} x +\operatorname {a2} \right ) y+x \left (\operatorname {b1} x +\operatorname {a1} \right ) y^{\prime }+x^{3} y^{\prime \prime }&=0 \end{align*}
✓ Maple. Time used: 0.140 (sec). Leaf size: 132
ode:=(b2*x+a2)*y(x)+x*(b1*x+a1)*diff(y(x),x)+x^3*diff(diff(y(x),x),x) = 0;
dsolve(ode,y(x), singsol=all);
\[
y = x^{-\frac {\operatorname {b1}}{2}+\frac {1}{2}-\frac {\sqrt {\operatorname {b1}^{2}-2 \operatorname {b1} -4 \operatorname {b2} +1}}{2}} \left (\operatorname {KummerU}\left (\frac {\sqrt {\operatorname {b1}^{2}-2 \operatorname {b1} -4 \operatorname {b2} +1}\, \operatorname {a1} +\left (-1+\operatorname {b1} \right ) \operatorname {a1} -2 \operatorname {a2}}{2 \operatorname {a1}}, 1+\sqrt {\operatorname {b1}^{2}-2 \operatorname {b1} -4 \operatorname {b2} +1}, \frac {\operatorname {a1}}{x}\right ) c_2 +\operatorname {KummerM}\left (\frac {\sqrt {\operatorname {b1}^{2}-2 \operatorname {b1} -4 \operatorname {b2} +1}\, \operatorname {a1} +\left (-1+\operatorname {b1} \right ) \operatorname {a1} -2 \operatorname {a2}}{2 \operatorname {a1}}, 1+\sqrt {\operatorname {b1}^{2}-2 \operatorname {b1} -4 \operatorname {b2} +1}, \frac {\operatorname {a1}}{x}\right ) c_1 \right )
\]
✓ Mathematica. Time used: 0.199 (sec). Leaf size: 255
ode=(a2 + b2*x)*y[x] + x*(a1 + b1*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^{-\sqrt {\text {b1}^2-2 \text {b1}-4 \text {b2}+1}+\text {b1}+1} \text {a1}^{\frac {1}{2} \left (-\sqrt {\text {b1}^2-2 \text {b1}-4 \text {b2}+1}+\text {b1}-1\right )} \left (\frac {1}{x}\right )^{\frac {1}{2} \left (-\sqrt {\text {b1}^2-2 \text {b1}-4 \text {b2}+1}+\text {b1}-1\right )} \left (c_2 i^{2 \sqrt {\text {b1}^2-2 \text {b1}-4 \text {b2}+1}} \text {a1}^{\sqrt {\text {b1}^2-2 \text {b1}-4 \text {b2}+1}} \left (\frac {1}{x}\right )^{\sqrt {\text {b1}^2-2 \text {b1}-4 \text {b2}+1}} \operatorname {Hypergeometric1F1}\left (\frac {1}{2} \left (-\frac {2 \text {a2}}{\text {a1}}+\text {b1}+\sqrt {\text {b1}^2-2 \text {b1}-4 \text {b2}+1}-1\right ),\sqrt {\text {b1}^2-2 \text {b1}-4 \text {b2}+1}+1,\frac {\text {a1}}{x}\right )+c_1 \operatorname {Hypergeometric1F1}\left (\frac {1}{2} \left (-\frac {2 \text {a2}}{\text {a1}}+\text {b1}-\sqrt {\text {b1}^2-2 \text {b1}-4 \text {b2}+1}-1\right ),1-\sqrt {\text {b1}^2-2 \text {b1}-4 \text {b2}+1},\frac {\text {a1}}{x}\right )\right ) \end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
a1 = symbols("a1")
a2 = symbols("a2")
b1 = symbols("b1")
b2 = symbols("b2")
y = Function("y")
ode = Eq(x**3*Derivative(y(x), (x, 2)) + x*(a1 + b1*x)*Derivative(y(x), x) + (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) - b2*x*y(x) - x**3*Derivative(y(