23.3.505 problem 511
Internal
problem
ID
[6219]
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
:
511
Date
solved
:
Friday, October 03, 2025 at 01:57:03 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 \left (x^{2}+\operatorname {a0} \right ) y^{\prime \prime }&=0 \end{align*}
✓ Maple. Time used: 0.033 (sec). Leaf size: 171
ode:=a2*x*y(x)+(b1*x^2+a1)*diff(y(x),x)+x*(x^2+a0)*diff(diff(y(x),x),x) = 0;
dsolve(ode,y(x), singsol=all);
\[
y = \left (x^{2}+\operatorname {a0} \right )^{\frac {\left (-\operatorname {b1} +2\right ) \operatorname {a0} +\operatorname {a1}}{2 \operatorname {a0}}} \left (x^{\frac {\operatorname {a0} -\operatorname {a1}}{\operatorname {a0}}} c_1 \operatorname {hypergeom}\left (\left [-\frac {\operatorname {b1}}{4}+\frac {5}{4}-\frac {\sqrt {\operatorname {b1}^{2}-4 \operatorname {a2} -2 \operatorname {b1} +1}}{4}, -\frac {\operatorname {b1}}{4}+\frac {5}{4}+\frac {\sqrt {\operatorname {b1}^{2}-4 \operatorname {a2} -2 \operatorname {b1} +1}}{4}\right ], \left [\frac {3}{2}-\frac {\operatorname {a1}}{2 \operatorname {a0}}\right ], -\frac {x^{2}}{\operatorname {a0}}\right )+\operatorname {hypergeom}\left (\left [-\frac {\sqrt {\operatorname {b1}^{2}-4 \operatorname {a2} -2 \operatorname {b1} +1}}{4}-\frac {\operatorname {b1}}{4}+\frac {3}{4}+\frac {\operatorname {a1}}{2 \operatorname {a0}}, -\frac {\operatorname {b1}}{4}+\frac {3}{4}+\frac {\operatorname {a1}}{2 \operatorname {a0}}+\frac {\sqrt {\operatorname {b1}^{2}-4 \operatorname {a2} -2 \operatorname {b1} +1}}{4}\right ], \left [\frac {1}{2}+\frac {\operatorname {a1}}{2 \operatorname {a0}}\right ], -\frac {x^{2}}{\operatorname {a0}}\right ) c_2 \right )
\]
✓ Mathematica. Time used: 0.338 (sec). Leaf size: 177
ode=a2*x*y[x] + (a1 + b1*x^2)*D[y[x],x] + x*(a0 + x^2)*D[y[x],{x,2}] == 0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to c_2 \text {a0}^{\frac {1}{2} \left (\frac {\text {a1}}{\text {a0}}-1\right )} x^{1-\frac {\text {a1}}{\text {a0}}} \operatorname {Hypergeometric2F1}\left (\frac {\text {a0} \left (\text {b1}-\sqrt {(\text {b1}-1)^2-4 \text {a2}}+1\right )-2 \text {a1}}{4 \text {a0}},\frac {\text {a0} \left (\text {b1}+\sqrt {(\text {b1}-1)^2-4 \text {a2}}+1\right )-2 \text {a1}}{4 \text {a0}},\frac {3}{2}-\frac {\text {a1}}{2 \text {a0}},-\frac {x^2}{\text {a0}}\right )+c_1 \operatorname {Hypergeometric2F1}\left (\frac {1}{4} \left (\text {b1}-\sqrt {(\text {b1}-1)^2-4 \text {a2}}-1\right ),\frac {1}{4} \left (\text {b1}+\sqrt {(\text {b1}-1)^2-4 \text {a2}}-1\right ),\frac {\text {a0}+\text {a1}}{2 \text {a0}},-\frac {x^2}{\text {a0}}\right ) \end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
a0 = symbols("a0")
a1 = symbols("a1")
a2 = symbols("a2")
b1 = symbols("b1")
y = Function("y")
ode = Eq(a2*x*y(x) + x*(a0 + x**2)*Derivative(y(x), (x, 2)) + (a1 + b1*x**2)*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
ValueError : Expected Expr or iterable but got None