23.3.487 problem 493
Internal
problem
ID
[6201]
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
:
493
Date
solved
:
Friday, October 03, 2025 at 01:56:53 AM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries]]
\begin{align*} \operatorname {a2} x y+\left (\operatorname {b1} \,x^{2}+\operatorname {a1} \right ) y^{\prime }+x^{3} y^{\prime \prime }&=0 \end{align*}
✓ Maple. Time used: 0.122 (sec). Leaf size: 120
ode:=a2*x*y(x)+(b1*x^2+a1)*diff(y(x),x)+x^3*diff(diff(y(x),x),x) = 0;
dsolve(ode,y(x), singsol=all);
\[
y = x^{-\frac {\operatorname {b1}}{2}+\frac {1}{2}-\frac {\sqrt {\operatorname {b1}^{2}-4 \operatorname {a2} -2 \operatorname {b1} +1}}{2}} \left (\operatorname {KummerU}\left (-\frac {1}{4}+\frac {\sqrt {\operatorname {b1}^{2}-4 \operatorname {a2} -2 \operatorname {b1} +1}}{4}+\frac {\operatorname {b1}}{4}, 1+\frac {\sqrt {\operatorname {b1}^{2}-4 \operatorname {a2} -2 \operatorname {b1} +1}}{2}, \frac {\operatorname {a1}}{2 x^{2}}\right ) c_2 +\operatorname {KummerM}\left (-\frac {1}{4}+\frac {\sqrt {\operatorname {b1}^{2}-4 \operatorname {a2} -2 \operatorname {b1} +1}}{4}+\frac {\operatorname {b1}}{4}, 1+\frac {\sqrt {\operatorname {b1}^{2}-4 \operatorname {a2} -2 \operatorname {b1} +1}}{2}, \frac {\operatorname {a1}}{2 x^{2}}\right ) c_1 \right )
\]
✓ Mathematica. Time used: 0.193 (sec). Leaf size: 284
ode=a2*x*y[x] + (a1 + b1*x^2)*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 -(-1)^{\frac {1}{4} \left (-\sqrt {(\text {b1}-1)^2-4 \text {a2}}+\text {b1}+3\right )} 2^{\frac {1}{4} \left (-\sqrt {(\text {b1}-1)^2-4 \text {a2}}-\text {b1}+1\right )} \text {a1}^{\frac {1}{4} \left (-\sqrt {(\text {b1}-1)^2-4 \text {a2}}+\text {b1}-1\right )} \left (\frac {1}{x}\right )^{\frac {1}{2} \left (-\sqrt {(\text {b1}-1)^2-4 \text {a2}}+\text {b1}-1\right )} \left (c_2 i^{\sqrt {(\text {b1}-1)^2-4 \text {a2}}} \text {a1}^{\frac {1}{2} \sqrt {(\text {b1}-1)^2-4 \text {a2}}} \left (\frac {1}{x}\right )^{\sqrt {(\text {b1}-1)^2-4 \text {a2}}} \operatorname {Hypergeometric1F1}\left (\frac {1}{4} \left (\text {b1}+\sqrt {(\text {b1}-1)^2-4 \text {a2}}-1\right ),\frac {1}{2} \left (\sqrt {(\text {b1}-1)^2-4 \text {a2}}+2\right ),\frac {\text {a1}}{2 x^2}\right )+c_1 2^{\frac {1}{2} \sqrt {(\text {b1}-1)^2-4 \text {a2}}} \operatorname {Hypergeometric1F1}\left (\frac {1}{4} \left (\text {b1}-\sqrt {(\text {b1}-1)^2-4 \text {a2}}-1\right ),1-\frac {1}{2} \sqrt {(\text {b1}-1)^2-4 \text {a2}},\frac {\text {a1}}{2 x^2}\right )\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*x*y(x) + x**3*Derivative(y(x), (x, 2)) + (a1 + b1*x**2)*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE -x*(-a2*y(x) - x**2*Derivative(y(x), (x, 2)))/(a1 + b1*x**2) + D