61.27.34 problem 44
Internal
problem
ID
[12465]
Book
:
Handbook
of
exact
solutions
for
ordinary
differential
equations.
By
Polyanin
and
Zaitsev.
Second
edition
Section
:
Chapter
2,
Second-Order
Differential
Equations.
section
2.1.2-2
Problem
number
:
44
Date
solved
:
Wednesday, March 05, 2025 at 07:07:06 PM
CAS
classification
:
[[_2nd_order, _missing_y]]
\begin{align*} y^{\prime \prime }+a \,x^{n} y^{\prime }&=0 \end{align*}
✓ Maple. Time used: 0.002 (sec). Leaf size: 203
ode:=diff(diff(y(x),x),x)+a*x^n*diff(y(x),x) = 0;
dsolve(ode,y(x), singsol=all);
\[
y = \frac {x^{-n} \left (c_{2} {\mathrm e}^{-\frac {x^{n} a x}{2 n +2}} \left (\frac {a \,x^{n} x}{n +1}\right )^{\frac {-n -2}{2 n +2}} \left (n +2\right )^{2} \left (n +1\right )^{2} \operatorname {WhittakerM}\left (\frac {n +2}{2 n +2}, \frac {2 n +3}{2 n +2}, \frac {a \,x^{n} x}{n +1}\right )+c_{2} {\mathrm e}^{-\frac {x^{n} a x}{2 n +2}} \left (\frac {a \,x^{n} x}{n +1}\right )^{\frac {-n -2}{2 n +2}} \left (n +1\right )^{3} \left (x^{n} a x +n +2\right ) \operatorname {WhittakerM}\left (-\frac {n}{2 n +2}, \frac {2 n +3}{2 n +2}, \frac {a \,x^{n} x}{n +1}\right )+c_{1} x^{n} \left (n +2\right )\right )}{n +2}
\]
✓ Mathematica. Time used: 0.037 (sec). Leaf size: 56
ode=D[y[x],{x,2}]+a*x^n*D[y[x],x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
y(x)\to c_2-\frac {c_1 x \left (\frac {a x^{n+1}}{n+1}\right )^{-\frac {1}{n+1}} \Gamma \left (\frac {1}{n+1},\frac {a x^{n+1}}{n+1}\right )}{n+1}
\]
✓ Sympy. Time used: 1.510 (sec). Leaf size: 258
from sympy import *
x = symbols("x")
a = symbols("a")
n = symbols("n")
y = Function("y")
ode = Eq(a*x**n*Derivative(y(x), x) + Derivative(y(x), (x, 2)),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
y{\left (x \right )} = \begin {cases} - \frac {C_{1} n \Gamma \left (\frac {n}{n + 1} + \frac {2}{n + 1}\right ) \gamma \left (\frac {1}{n + 1}, \frac {a x^{n + 1}}{\operatorname {polar\_lift}{\left (n + 1 \right )}}\right ) \operatorname {polar\_lift}^{\frac {1}{n + 1}}{\left (n + 1 \right )}}{a^{\frac {1}{n + 1}} n^{2} \Gamma \left (\frac {2 n}{n + 1} + \frac {3}{n + 1}\right ) + 2 a^{\frac {1}{n + 1}} n \Gamma \left (\frac {2 n}{n + 1} + \frac {3}{n + 1}\right ) + a^{\frac {1}{n + 1}} \Gamma \left (\frac {2 n}{n + 1} + \frac {3}{n + 1}\right )} - \frac {2 C_{1} \Gamma \left (\frac {n}{n + 1} + \frac {2}{n + 1}\right ) \gamma \left (\frac {1}{n + 1}, \frac {a x^{n + 1}}{\operatorname {polar\_lift}{\left (n + 1 \right )}}\right ) \operatorname {polar\_lift}^{\frac {1}{n + 1}}{\left (n + 1 \right )}}{a^{\frac {1}{n + 1}} n^{2} \Gamma \left (\frac {2 n}{n + 1} + \frac {3}{n + 1}\right ) + 2 a^{\frac {1}{n + 1}} n \Gamma \left (\frac {2 n}{n + 1} + \frac {3}{n + 1}\right ) + a^{\frac {1}{n + 1}} \Gamma \left (\frac {2 n}{n + 1} + \frac {3}{n + 1}\right )} - C_{2} & \text {for}\: n > -\infty \wedge n < \infty \wedge n \neq -1 \\\frac {C_{1} x}{a e^{a \log {\left (x \right )}} - e^{a \log {\left (x \right )}}} - C_{2} & \text {for}\: a \neq 1 \\- C_{1} \log {\left (x \right )} - C_{2} & \text {otherwise} \end {cases}
\]