23.3.488 problem 494
Internal
problem
ID
[6202]
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
:
494
Date
solved
:
Friday, October 03, 2025 at 01:56:54 AM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries]]
\begin{align*} \operatorname {a2} y+x \left (\operatorname {b1} x +\operatorname {a1} \right ) y^{\prime }+x^{3} y^{\prime \prime }&=0 \end{align*}
✓ Maple. Time used: 0.043 (sec). Leaf size: 146
ode:=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);
\[
y = \frac {x^{-\operatorname {b1}} \left (-x \operatorname {a2} c_2 \left (\operatorname {b1} \operatorname {a1} -\operatorname {a2} \right ) \left (\operatorname {a1} -\operatorname {a2} \right ) \operatorname {KummerU}\left (\frac {\left (\operatorname {b1} +1\right ) \operatorname {a1} -\operatorname {a2}}{\operatorname {a1}}, \operatorname {b1} , \frac {\operatorname {a1}}{x}\right )+\left (x \operatorname {a1} c_1 \left (\operatorname {b1} \operatorname {a1} -\operatorname {a2} \right ) \operatorname {KummerM}\left (\frac {\left (\operatorname {b1} +1\right ) \operatorname {a1} -\operatorname {a2}}{\operatorname {a1}}, \operatorname {b1} , \frac {\operatorname {a1}}{x}\right )-\left (\operatorname {a1} c_1 \operatorname {KummerM}\left (\frac {\operatorname {b1} \operatorname {a1} -\operatorname {a2}}{\operatorname {a1}}, \operatorname {b1} , \frac {\operatorname {a1}}{x}\right )-\operatorname {a2} c_2 \operatorname {KummerU}\left (\frac {\operatorname {b1} \operatorname {a1} -\operatorname {a2}}{\operatorname {a1}}, \operatorname {b1} , \frac {\operatorname {a1}}{x}\right )\right ) \left (\operatorname {b1} \operatorname {a1} x +\operatorname {a1}^{2}-2 \operatorname {a2} x \right )\right ) \operatorname {a1} \right )}{\operatorname {a1}^{2} \operatorname {a2}}
\]
✓ Mathematica. Time used: 0.157 (sec). Leaf size: 61
ode=a2*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 c_1 \operatorname {Hypergeometric1F1}\left (-\frac {\text {a2}}{\text {a1}},2-\text {b1},\frac {\text {a1}}{x}\right )-(-1)^{\text {b1}} c_2 \text {a1}^{\text {b1}-1} \left (\frac {1}{x}\right )^{\text {b1}-1} \operatorname {Hypergeometric1F1}\left (-\frac {\text {a2}}{\text {a1}}+\text {b1}-1,\text {b1},\frac {\text {a1}}{x}\right ) \end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
a1 = symbols("a1")
a2 = symbols("a2")
b1 = symbols("b1")
y = Function("y")
ode = Eq(a2*y(x) + x**3*Derivative(y(x), (x, 2)) + x*(a1 + b1*x)*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE Derivative(y(x), x) - (-a2*y(x) - x**3*Derivative(y(x), (x, 2)))