23.3.586 problem 594
Internal
problem
ID
[6300]
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
:
594
Date
solved
:
Friday, October 03, 2025 at 02:00:25 AM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries]]
\begin{align*} \left (\operatorname {a2} +\operatorname {b2} \,x^{k}\right ) y+x \left (\operatorname {a1} +\operatorname {b1} \,x^{k}\right ) y^{\prime }+x^{2} \left (\operatorname {a0} +\operatorname {b0} \,x^{k}\right ) y^{\prime \prime }&=0 \end{align*}
✓ Maple. Time used: 0.074 (sec). Leaf size: 391
ode:=(a2+b2*x^k)*y(x)+x*(a1+b1*x^k)*diff(y(x),x)+x^2*(a0+b0*x^k)*diff(diff(y(x),x),x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*} \text {Solution too large to show}\end{align*}
✓ Mathematica. Time used: 0.205 (sec). Leaf size: 540
ode=(a2 + b2*x^k)*y[x] + x*(a1 + b1*x^k)*D[y[x],x] + x^2*(a0 + b0*x^k)*D[y[x],{x,2}] == 0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \text {a0}^{-\frac {\sqrt {\text {a0}^2-2 \text {a0} (\text {a1}+2 \text {a2})+\text {a1}^2}+\text {a0}-\text {a1}}{2 \text {a0} k}} \text {b0}^{-\frac {\sqrt {\text {a0}^2-2 \text {a0} (\text {a1}+2 \text {a2})+\text {a1}^2}-\text {a0}+\text {a1}}{2 \text {a0} k}} \left (x^k\right )^{-\frac {\sqrt {\text {a0}^2-2 \text {a0} (\text {a1}+2 \text {a2})+\text {a1}^2}-\text {a0}+\text {a1}}{2 \text {a0} k}} \left (c_2 \text {b0}^{\frac {\sqrt {\text {a0}^2-2 \text {a0} (\text {a1}+2 \text {a2})+\text {a1}^2}}{\text {a0} k}} \left (x^k\right )^{\frac {\sqrt {\text {a0}^2-2 \text {a0} (\text {a1}+2 \text {a2})+\text {a1}^2}}{\text {a0} k}} \operatorname {Hypergeometric2F1}\left (\frac {\text {b1} \text {a0}-\sqrt {\text {b0}^2-2 (\text {b1}+2 \text {b2}) \text {b0}+\text {b1}^2} \text {a0}-\text {a1} \text {b0}+\sqrt {\text {a0}^2-2 (\text {a1}+2 \text {a2}) \text {a0}+\text {a1}^2} \text {b0}}{2 \text {a0} \text {b0} k},\frac {\text {b1} \text {a0}+\sqrt {\text {b0}^2-2 (\text {b1}+2 \text {b2}) \text {b0}+\text {b1}^2} \text {a0}-\text {a1} \text {b0}+\sqrt {\text {a0}^2-2 (\text {a1}+2 \text {a2}) \text {a0}+\text {a1}^2} \text {b0}}{2 \text {a0} \text {b0} k},\frac {\sqrt {\text {a0}^2-2 (\text {a1}+2 \text {a2}) \text {a0}+\text {a1}^2}}{\text {a0} k}+1,-\frac {\text {b0} x^k}{\text {a0}}\right )+c_1 \text {a0}^{\frac {\sqrt {\text {a0}^2-2 \text {a0} (\text {a1}+2 \text {a2})+\text {a1}^2}}{\text {a0} k}} \operatorname {Hypergeometric2F1}\left (-\frac {-\text {b1} \text {a0}+\sqrt {\text {b0}^2-2 (\text {b1}+2 \text {b2}) \text {b0}+\text {b1}^2} \text {a0}+\text {a1} \text {b0}+\sqrt {\text {a0}^2-2 (\text {a1}+2 \text {a2}) \text {a0}+\text {a1}^2} \text {b0}}{2 \text {a0} \text {b0} k},\frac {\text {b1} \text {a0}+\sqrt {\text {b0}^2-2 (\text {b1}+2 \text {b2}) \text {b0}+\text {b1}^2} \text {a0}-\text {a1} \text {b0}-\sqrt {\text {a0}^2-2 (\text {a1}+2 \text {a2}) \text {a0}+\text {a1}^2} \text {b0}}{2 \text {a0} \text {b0} k},1-\frac {\sqrt {\text {a0}^2-2 (\text {a1}+2 \text {a2}) \text {a0}+\text {a1}^2}}{\text {a0} k},-\frac {\text {b0} x^k}{\text {a0}}\right )\right ) \end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
a0 = symbols("a0")
a1 = symbols("a1")
a2 = symbols("a2")
b0 = symbols("b0")
b1 = symbols("b1")
b2 = symbols("b2")
k = symbols("k")
y = Function("y")
ode = Eq(x**2*(a0 + b0*x**k)*Derivative(y(x), (x, 2)) + x*(a1 + b1*x**k)*Derivative(y(x), x) + (a2 + b2*x**k)*y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
TypeError : Symbol object cannot be interpreted as an integer