61.29.9 problem 118
Internal
problem
ID
[12539]
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
:
118
Date
solved
:
Monday, March 31, 2025 at 05:38:22 AM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries]]
\begin{align*} x^{2} y^{\prime \prime }+\left (a \,x^{n}+b \right ) y&=0 \end{align*}
✓ Maple. Time used: 0.014 (sec). Leaf size: 63
ode:=x^2*diff(diff(y(x),x),x)+(a*x^n+b)*y(x) = 0;
dsolve(ode,y(x), singsol=all);
\[
y = \left (\operatorname {BesselY}\left (\frac {\sqrt {1-4 b}}{n}, \frac {2 \sqrt {a}\, x^{\frac {n}{2}}}{n}\right ) c_2 +\operatorname {BesselJ}\left (\frac {\sqrt {1-4 b}}{n}, \frac {2 \sqrt {a}\, x^{\frac {n}{2}}}{n}\right ) c_1 \right ) \sqrt {x}
\]
✓ Mathematica. Time used: 0.208 (sec). Leaf size: 351
ode=x^2*D[y[x],{x,2}]+(a*x^n+b)*y[x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
y(x)\to n^{-\frac {\sqrt {(1-4 b) n^2}+i \sqrt {4 b-1} n+n}{n^2}} a^{\frac {-\sqrt {(1-4 b) n^2}-i \sqrt {4 b-1} n+n}{2 n^2}} \left (x^n\right )^{\frac {-\sqrt {(1-4 b) n^2}-i \sqrt {4 b-1} n+n}{2 n^2}} \left (c_2 n^{\frac {2 \sqrt {(1-4 b) n^2}}{n^2}} a^{\frac {i \sqrt {4 b-1}}{n}} \left (x^n\right )^{\frac {i \sqrt {4 b-1}}{n}} \operatorname {Gamma}\left (\frac {n+\sqrt {1-4 b}}{n}\right ) \operatorname {BesselJ}\left (\frac {\sqrt {(1-4 b) n^2}}{n^2},\frac {2 \sqrt {a} \sqrt {x^n}}{n}\right )+c_1 n^{\frac {2 i \sqrt {4 b-1}}{n}} a^{\frac {\sqrt {(1-4 b) n^2}}{n^2}} \left (x^n\right )^{\frac {\sqrt {(1-4 b) n^2}}{n^2}} \operatorname {Gamma}\left (1-\frac {\sqrt {1-4 b}}{n}\right ) \operatorname {BesselJ}\left (-\frac {\sqrt {(1-4 b) n^2}}{n^2},\frac {2 \sqrt {a} \sqrt {x^n}}{n}\right )\right )
\]
✓ Sympy. Time used: 0.130 (sec). Leaf size: 61
from sympy import *
x = symbols("x")
a = symbols("a")
b = symbols("b")
n = symbols("n")
y = Function("y")
ode = Eq(x**2*Derivative(y(x), (x, 2)) + (a*x**n + b)*y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
y{\left (x \right )} = \sqrt {x} \left (C_{1} J_{\frac {2 \sqrt {\frac {1}{4} - b}}{n}}\left (\frac {2 \sqrt {a} x^{\frac {n}{2}}}{n}\right ) + C_{2} Y_{\frac {2 \sqrt {\frac {1}{4} - b}}{n}}\left (\frac {2 \sqrt {a} x^{\frac {n}{2}}}{n}\right )\right )
\]