61.29.37 problem 146
Internal
problem
ID
[12567]
Book
:
Handbook
of
exact
solutions
for
ordinary
differential
equations.
By
Polyanin
and
Zaitsev.
Second
edition
Section
:
Chapter
2,
Second-Order
Differential
Equations.
section
2.1.2-4
Problem
number
:
146
Date
solved
:
Thursday, March 13, 2025 at 11:43:48 PM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries]]
\begin{align*} x^{2} y^{\prime \prime }+x \left (a \,x^{n}+b \right ) y^{\prime }+\left (\alpha \,x^{2 n}+\beta \,x^{n}+\gamma \right ) y&=0 \end{align*}
✓ Maple. Time used: 0.755 (sec). Leaf size: 148
ode:=x^2*diff(diff(y(x),x),x)+x*(a*x^n+b)*diff(y(x),x)+(alpha*x^(2*n)+beta*x^n+gamma)*y(x) = 0;
dsolve(ode,y(x), singsol=all);
\[
y = x^{-\frac {b}{2}-\frac {n}{2}+\frac {1}{2}} {\mathrm e}^{-\frac {a \,x^{n}}{2 n}} \left (c_{1} \operatorname {WhittakerM}\left (-\frac {a \left (b +n -1\right )-2 \beta }{2 \sqrt {a^{2}-4 \alpha }\, n}, \frac {\sqrt {b^{2}-2 b -4 \gamma +1}}{2 n}, \frac {\sqrt {a^{2}-4 \alpha }\, x^{n}}{n}\right )+c_{2} \operatorname {WhittakerW}\left (-\frac {a \left (b +n -1\right )-2 \beta }{2 \sqrt {a^{2}-4 \alpha }\, n}, \frac {\sqrt {b^{2}-2 b -4 \gamma +1}}{2 n}, \frac {\sqrt {a^{2}-4 \alpha }\, x^{n}}{n}\right )\right )
\]
✓ Mathematica. Time used: 0.311 (sec). Leaf size: 420
ode=x^2*D[y[x],{x,2}]+x*(a*x^n+b)*D[y[x],x]+(\[Alpha]*x^(2*n)+\[Beta]*x^n+\[Gamma])*y[x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
y(x)\to x^{\frac {1}{2}-\frac {n}{2}} 2^{\frac {1}{2} \left (\frac {\sqrt {n^2 \left (b^2-2 b-4 \gamma +1\right )}}{n^2}+1\right )} e^{-\frac {\left (\sqrt {a^2-4 \alpha }+a\right ) x^n}{2 n}} \left (x^n\right )^{\frac {\sqrt {n^2 \left (b^2-2 b-4 \gamma +1\right )}-b n+n^2}{2 n^2}} \left (c_1 \operatorname {HypergeometricU}\left (\frac {\left (n^2+\sqrt {n^2 \left (b^2-2 b-4 \gamma +1\right )}\right ) a^2+n (b+n-1) \sqrt {a^2-4 \alpha } a-2 \left (2 \alpha n^2+\sqrt {a^2-4 \alpha } \beta n+2 \alpha \sqrt {n^2 \left (b^2-2 b-4 \gamma +1\right )}\right )}{2 n^2 \left (a^2-4 \alpha \right )},\frac {n^2+\sqrt {n^2 \left (b^2-2 b-4 \gamma +1\right )}}{n^2},\frac {x^n \sqrt {a^2-4 \alpha }}{n}\right )+c_2 L_{\frac {-\left (\left (n^2+\sqrt {n^2 \left (b^2-2 b-4 \gamma +1\right )}\right ) a^2\right )-n (b+n-1) \sqrt {a^2-4 \alpha } a+4 n^2 \alpha +2 n \sqrt {a^2-4 \alpha } \beta +4 \alpha \sqrt {n^2 \left (b^2-2 b-4 \gamma +1\right )}}{2 n^2 \left (a^2-4 \alpha \right )}}^{\frac {\sqrt {n^2 \left (b^2-2 b-4 \gamma +1\right )}}{n^2}}\left (\frac {x^n \sqrt {a^2-4 \alpha }}{n}\right )\right )
\]
✗ Sympy
from sympy import *
x = symbols("x")
Alpha = symbols("Alpha")
BETA = symbols("BETA")
Gamma = symbols("Gamma")
a = symbols("a")
b = symbols("b")
n = symbols("n")
y = Function("y")
ode = Eq(x**2*Derivative(y(x), (x, 2)) + x*(a*x**n + b)*Derivative(y(x), x) + (Alpha*x**(2*n) + BETA*x**n + Gamma)*y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
ValueError : Expected Expr or iterable but got None