23.3.315 problem 317
Internal
problem
ID
[6029]
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
:
317
Date
solved
:
Tuesday, September 30, 2025 at 02:20:09 PM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries]]
\begin{align*} \left (b +c \,x^{2 k}\right ) y+a x y^{\prime }+x^{2} y^{\prime \prime }&=0 \end{align*}
✓ Maple. Time used: 0.012 (sec). Leaf size: 76
ode:=(b+c*x^(2*k))*y(x)+a*x*diff(y(x),x)+x^2*diff(diff(y(x),x),x) = 0;
dsolve(ode,y(x), singsol=all);
\[
y = x^{-\frac {a}{2}} \sqrt {x}\, \left (\operatorname {BesselY}\left (\frac {\sqrt {a^{2}-2 a -4 b +1}}{2 k}, \frac {\sqrt {c}\, x^{k}}{k}\right ) c_2 +\operatorname {BesselJ}\left (\frac {\sqrt {a^{2}-2 a -4 b +1}}{2 k}, \frac {\sqrt {c}\, x^{k}}{k}\right ) c_1 \right )
\]
✓ Mathematica. Time used: 0.179 (sec). Leaf size: 561
ode=(b + c*x^(2*k))*y[x] + a*x*D[y[x],x] + x^2*D[y[x],{x,2}] == 0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to 2^{-\frac {\sqrt {k^2 \left (a^2-2 a-4 b+1\right )}+k \sqrt {a^2-2 a-4 b+1}-a k+k}{2 k^2}} k^{-\frac {\sqrt {k^2 \left (a^2-2 a-4 b+1\right )}+k \sqrt {a^2-2 a-4 b+1}-a k+k}{2 k^2}} c^{-\frac {\sqrt {k^2 \left (a^2-2 a-4 b+1\right )}+k \left (\sqrt {a^2-2 a-4 b+1}+a-1\right )}{4 k^2}} \left (x^{2 k}\right )^{-\frac {\sqrt {k^2 \left (a^2-2 a-4 b+1\right )}+k \left (\sqrt {a^2-2 a-4 b+1}+a-1\right )}{4 k^2}} \left (c_2 2^{\frac {\sqrt {k^2 \left (a^2-2 a-4 b+1\right )}}{k^2}} k^{\frac {\sqrt {k^2 \left (a^2-2 a-4 b+1\right )}}{k^2}} c^{\frac {\sqrt {a^2-2 a-4 b+1}}{2 k}} \left (x^{2 k}\right )^{\frac {\sqrt {a^2-2 a-4 b+1}}{2 k}} \operatorname {Gamma}\left (\frac {\sqrt {a^2-2 a-4 b+1}}{2 k}+1\right ) \operatorname {BesselJ}\left (\frac {\sqrt {\left (a^2-2 a-4 b+1\right ) k^2}}{2 k^2},\frac {\sqrt {c} \sqrt {x^{2 k}}}{k}\right )+c_1 2^{\frac {\sqrt {a^2-2 a-4 b+1}}{k}} k^{\frac {\sqrt {a^2-2 a-4 b+1}}{k}} c^{\frac {\sqrt {k^2 \left (a^2-2 a-4 b+1\right )}}{2 k^2}} \left (x^{2 k}\right )^{\frac {\sqrt {k^2 \left (a^2-2 a-4 b+1\right )}}{2 k^2}} \operatorname {Gamma}\left (1-\frac {\sqrt {a^2-2 a-4 b+1}}{2 k}\right ) \operatorname {BesselJ}\left (-\frac {\sqrt {\left (a^2-2 a-4 b+1\right ) k^2}}{2 k^2},\frac {\sqrt {c} \sqrt {x^{2 k}}}{k}\right )\right ) \end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
b = symbols("b")
c = symbols("c")
k = symbols("k")
y = Function("y")
ode = Eq(a*x*Derivative(y(x), x) + x**2*Derivative(y(x), (x, 2)) + (b + c*x**(2*k))*y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
TypeError : invalid input: 1 - a