23.3.336 problem 338
Internal
problem
ID
[6050]
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
:
338
Date
solved
:
Friday, October 03, 2025 at 01:45:57 AM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries]]
\begin{align*} \left (\operatorname {a1} +\operatorname {b1} \,x^{k}+\operatorname {c1} \,x^{2 k}\right ) y+x \left (\operatorname {a0} +\operatorname {b0} \,x^{k}\right ) y^{\prime }+x^{2} y^{\prime \prime }&=0 \end{align*}
✓ Maple. Time used: 0.047 (sec). Leaf size: 148
ode:=(a1+b1*x^k+c1*x^(2*k))*y(x)+x*(a0+b0*x^k)*diff(y(x),x)+x^2*diff(diff(y(x),x),x) = 0;
dsolve(ode,y(x), singsol=all);
\[
y = {\mathrm e}^{-\frac {\operatorname {b0} \,x^{k}}{2 k}} x^{\frac {1}{2}-\frac {k}{2}-\frac {\operatorname {a0}}{2}} \left (\operatorname {WhittakerM}\left (-\frac {\left (k +\operatorname {a0} -1\right ) \operatorname {b0} -2 \operatorname {b1}}{2 \sqrt {\operatorname {b0}^{2}-4 \operatorname {c1}}\, k}, \frac {\sqrt {\operatorname {a0}^{2}-2 \operatorname {a0} -4 \operatorname {a1} +1}}{2 k}, \frac {\sqrt {\operatorname {b0}^{2}-4 \operatorname {c1}}\, x^{k}}{k}\right ) c_1 +\operatorname {WhittakerW}\left (-\frac {\left (k +\operatorname {a0} -1\right ) \operatorname {b0} -2 \operatorname {b1}}{2 \sqrt {\operatorname {b0}^{2}-4 \operatorname {c1}}\, k}, \frac {\sqrt {\operatorname {a0}^{2}-2 \operatorname {a0} -4 \operatorname {a1} +1}}{2 k}, \frac {\sqrt {\operatorname {b0}^{2}-4 \operatorname {c1}}\, x^{k}}{k}\right ) c_2 \right )
\]
✓ Mathematica. Time used: 0.672 (sec). Leaf size: 412
ode=(a1 + b1*x^k + c1*x^(2*k))*y[x] + x*(a0 + b0*x^k)*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 x^{\frac {1}{2}-\frac {k}{2}} 2^{\frac {1}{2} \left (\frac {\sqrt {k^2 \left (\text {a0}^2-2 \text {a0}-4 \text {a1}+1\right )}}{k^2}+1\right )} \left (x^k\right )^{\frac {\sqrt {k^2 \left (\text {a0}^2-2 \text {a0}-4 \text {a1}+1\right )}-\text {a0} k+k^2}{2 k^2}} e^{-\frac {\left (\sqrt {\text {b0}^2-4 \text {c1}}+\text {b0}\right ) x^k}{2 k}} \left (c_1 \operatorname {HypergeometricU}\left (\frac {\left (k^2+\sqrt {\left (\text {a0}^2-2 \text {a0}-4 \text {a1}+1\right ) k^2}\right ) \text {b0}^2+\sqrt {\text {b0}^2-4 \text {c1}} k (\text {a0}+k-1) \text {b0}-2 \text {b1} \sqrt {\text {b0}^2-4 \text {c1}} k-4 \text {c1} \left (k^2+\sqrt {\left (\text {a0}^2-2 \text {a0}-4 \text {a1}+1\right ) k^2}\right )}{2 \left (\text {b0}^2-4 \text {c1}\right ) k^2},\frac {k^2+\sqrt {\left (\text {a0}^2-2 \text {a0}-4 \text {a1}+1\right ) k^2}}{k^2},\frac {\sqrt {\text {b0}^2-4 \text {c1}} x^k}{k}\right )+c_2 L_{-\frac {\left (k^2+\sqrt {\left (\text {a0}^2-2 \text {a0}-4 \text {a1}+1\right ) k^2}\right ) \text {b0}^2+\sqrt {\text {b0}^2-4 \text {c1}} k (\text {a0}+k-1) \text {b0}-2 \text {b1} \sqrt {\text {b0}^2-4 \text {c1}} k-4 \text {c1} \left (k^2+\sqrt {\left (\text {a0}^2-2 \text {a0}-4 \text {a1}+1\right ) k^2}\right )}{2 \left (\text {b0}^2-4 \text {c1}\right ) k^2}}^{\frac {\sqrt {\left (\text {a0}^2-2 \text {a0}-4 \text {a1}+1\right ) k^2}}{k^2}}\left (\frac {\sqrt {\text {b0}^2-4 \text {c1}} x^k}{k}\right )\right ) \end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
a0 = symbols("a0")
a1 = symbols("a1")
b0 = symbols("b0")
b1 = symbols("b1")
c1 = symbols("c1")
k = symbols("k")
y = Function("y")
ode = Eq(x**2*Derivative(y(x), (x, 2)) + x*(a0 + b0*x**k)*Derivative(y(x), x) + (a1 + b1*x**k + c1*x**(2*k))*y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
ValueError : Expected Expr or iterable but got None