60.1.30 problem 30
Internal
problem
ID
[10044]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
linear
first
order
Problem
number
:
30
Date
solved
:
Sunday, March 30, 2025 at 02:55:17 PM
CAS
classification
:
[_Riccati]
\begin{align*} y^{\prime }+x^{-a -1} y^{2}-x^{a}&=0 \end{align*}
✓ Maple. Time used: 0.002 (sec). Leaf size: 51
ode:=diff(y(x),x)+x^(-a-1)*y(x)^2-x^a = 0;
dsolve(ode,y(x), singsol=all);
\[
y = \frac {x^{\frac {1}{2}+a} \left (-\operatorname {BesselK}\left (a +1, 2 \sqrt {x}\right ) c_1 +\operatorname {BesselI}\left (a +1, 2 \sqrt {x}\right )\right )}{\operatorname {BesselK}\left (a , 2 \sqrt {x}\right ) c_1 +\operatorname {BesselI}\left (a , 2 \sqrt {x}\right )}
\]
✓ Mathematica. Time used: 0.396 (sec). Leaf size: 265
ode=D[y[x],x] + x^(-a-1)*y[x]^2 - x^a==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to \frac {x^a \left (\sqrt {x} \operatorname {Gamma}(1-a) \operatorname {BesselI}\left (-a-1,2 \sqrt {x}\right )+\sqrt {x} \operatorname {Gamma}(1-a) \operatorname {BesselI}\left (1-a,2 \sqrt {x}\right )-a \operatorname {Gamma}(1-a) \operatorname {BesselI}\left (-a,2 \sqrt {x}\right )+(-1)^a c_1 \sqrt {x} \operatorname {Gamma}(a+1) \operatorname {BesselI}\left (a-1,2 \sqrt {x}\right )-(-1)^a a c_1 \operatorname {Gamma}(a+1) \operatorname {BesselI}\left (a,2 \sqrt {x}\right )+(-1)^a c_1 \sqrt {x} \operatorname {Gamma}(a+1) \operatorname {BesselI}\left (a+1,2 \sqrt {x}\right )\right )}{2 \left (\operatorname {Gamma}(1-a) \operatorname {BesselI}\left (-a,2 \sqrt {x}\right )+(-1)^a c_1 \operatorname {Gamma}(a+1) \operatorname {BesselI}\left (a,2 \sqrt {x}\right )\right )} \\
y(x)\to \frac {x^a \left (\sqrt {x} \operatorname {BesselI}\left (a-1,2 \sqrt {x}\right )-a \operatorname {BesselI}\left (a,2 \sqrt {x}\right )+\sqrt {x} \operatorname {BesselI}\left (a+1,2 \sqrt {x}\right )\right )}{2 \operatorname {BesselI}\left (a,2 \sqrt {x}\right )} \\
\end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
y = Function("y")
ode = Eq(-x**a + x**(-a - 1)*y(x)**2 + Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE -x**a + x**(-a - 1)*y(x)**2 + Derivative(y(x), x) cannot be solved by the lie group method