56.1.38 problem 39
Internal
problem
ID
[8750]
Book
:
Own
collection
of
miscellaneous
problems
Section
:
section
1.0
Problem
number
:
39
Date
solved
:
Sunday, March 30, 2025 at 01:30:12 PM
CAS
classification
:
[_rational, _Riccati]
\begin{align*} u^{\prime }+u^{2}&=\frac {1}{x^{{4}/{5}}} \end{align*}
✓ Maple. Time used: 0.002 (sec). Leaf size: 46
ode:=diff(u(x),x)+u(x)^2 = 1/x^(4/5);
dsolve(ode,u(x), singsol=all);
\[
u = \frac {\operatorname {BesselI}\left (-\frac {1}{6}, \frac {5 x^{{3}/{5}}}{3}\right ) c_1 -\operatorname {BesselK}\left (\frac {1}{6}, \frac {5 x^{{3}/{5}}}{3}\right )}{x^{{2}/{5}} \left (c_1 \operatorname {BesselI}\left (\frac {5}{6}, \frac {5 x^{{3}/{5}}}{3}\right )+\operatorname {BesselK}\left (\frac {5}{6}, \frac {5 x^{{3}/{5}}}{3}\right )\right )}
\]
✓ Mathematica. Time used: 0.274 (sec). Leaf size: 286
ode=D[u[x],x]+u[x]^2==x^(-4/5);
ic={};
DSolve[{ode,ic},u[x],x,IncludeSingularSolutions->True]
\begin{align*}
u(x)\to \frac {(-1)^{5/6} x^{3/5} \operatorname {Gamma}\left (\frac {11}{6}\right ) \operatorname {BesselI}\left (-\frac {1}{6},\frac {5 x^{3/5}}{3}\right )+(-1)^{5/6} \operatorname {Gamma}\left (\frac {11}{6}\right ) \operatorname {BesselI}\left (\frac {5}{6},\frac {5 x^{3/5}}{3}\right )+(-1)^{5/6} x^{3/5} \operatorname {Gamma}\left (\frac {11}{6}\right ) \operatorname {BesselI}\left (\frac {11}{6},\frac {5 x^{3/5}}{3}\right )+c_1 x^{3/5} \operatorname {Gamma}\left (\frac {1}{6}\right ) \operatorname {BesselI}\left (-\frac {11}{6},\frac {5 x^{3/5}}{3}\right )+c_1 \operatorname {Gamma}\left (\frac {1}{6}\right ) \operatorname {BesselI}\left (-\frac {5}{6},\frac {5 x^{3/5}}{3}\right )+c_1 x^{3/5} \operatorname {Gamma}\left (\frac {1}{6}\right ) \operatorname {BesselI}\left (\frac {1}{6},\frac {5 x^{3/5}}{3}\right )}{2 x \left ((-1)^{5/6} \operatorname {Gamma}\left (\frac {11}{6}\right ) \operatorname {BesselI}\left (\frac {5}{6},\frac {5 x^{3/5}}{3}\right )+c_1 \operatorname {Gamma}\left (\frac {1}{6}\right ) \operatorname {BesselI}\left (-\frac {5}{6},\frac {5 x^{3/5}}{3}\right )\right )} \\
u(x)\to \frac {x^{3/5} \operatorname {BesselI}\left (-\frac {11}{6},\frac {5 x^{3/5}}{3}\right )+\operatorname {BesselI}\left (-\frac {5}{6},\frac {5 x^{3/5}}{3}\right )+x^{3/5} \operatorname {BesselI}\left (\frac {1}{6},\frac {5 x^{3/5}}{3}\right )}{2 x \operatorname {BesselI}\left (-\frac {5}{6},\frac {5 x^{3/5}}{3}\right )} \\
\end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
u = Function("u")
ode = Eq(u(x)**2 + Derivative(u(x), x) - 1/x**(4/5),0)
ics = {}
dsolve(ode,func=u(x),ics=ics)
NotImplementedError : The given ODE u(x)**2 + Derivative(u(x), x) - 1/x**(4/5) cannot be solved by the lie group method