29.3.4 problem 58
Internal
problem
ID
[4666]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
3
Problem
number
:
58
Date
solved
:
Sunday, March 30, 2025 at 03:35:29 AM
CAS
classification
:
[[_Riccati, _special]]
\begin{align*} y^{\prime }&=a \,x^{2}+b y^{2} \end{align*}
✓ Maple. Time used: 0.003 (sec). Leaf size: 71
ode:=diff(y(x),x) = x^2*a+b*y(x)^2;
dsolve(ode,y(x), singsol=all);
\[
y = -\frac {\sqrt {a b}\, x \left (\operatorname {BesselJ}\left (-\frac {3}{4}, \frac {\sqrt {a b}\, x^{2}}{2}\right ) c_1 +\operatorname {BesselY}\left (-\frac {3}{4}, \frac {\sqrt {a b}\, x^{2}}{2}\right )\right )}{b \left (c_1 \operatorname {BesselJ}\left (\frac {1}{4}, \frac {\sqrt {a b}\, x^{2}}{2}\right )+\operatorname {BesselY}\left (\frac {1}{4}, \frac {\sqrt {a b}\, x^{2}}{2}\right )\right )}
\]
✓ Mathematica. Time used: 0.171 (sec). Leaf size: 305
ode=D[y[x],x]==a x^2+b y[x]^2;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to \frac {\sqrt {a} \sqrt {b} x^2 \left (-2 \operatorname {BesselJ}\left (-\frac {3}{4},\frac {1}{2} \sqrt {a} \sqrt {b} x^2\right )+c_1 \left (\operatorname {BesselJ}\left (\frac {3}{4},\frac {1}{2} \sqrt {a} \sqrt {b} x^2\right )-\operatorname {BesselJ}\left (-\frac {5}{4},\frac {1}{2} \sqrt {a} \sqrt {b} x^2\right )\right )\right )-c_1 \operatorname {BesselJ}\left (-\frac {1}{4},\frac {1}{2} \sqrt {a} \sqrt {b} x^2\right )}{2 b x \left (\operatorname {BesselJ}\left (\frac {1}{4},\frac {1}{2} \sqrt {a} \sqrt {b} x^2\right )+c_1 \operatorname {BesselJ}\left (-\frac {1}{4},\frac {1}{2} \sqrt {a} \sqrt {b} x^2\right )\right )} \\
y(x)\to -\frac {\sqrt {a} \sqrt {b} x^2 \operatorname {BesselJ}\left (-\frac {5}{4},\frac {1}{2} \sqrt {a} \sqrt {b} x^2\right )-\sqrt {a} \sqrt {b} x^2 \operatorname {BesselJ}\left (\frac {3}{4},\frac {1}{2} \sqrt {a} \sqrt {b} x^2\right )+\operatorname {BesselJ}\left (-\frac {1}{4},\frac {1}{2} \sqrt {a} \sqrt {b} x^2\right )}{2 b x \operatorname {BesselJ}\left (-\frac {1}{4},\frac {1}{2} \sqrt {a} \sqrt {b} x^2\right )} \\
\end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
b = symbols("b")
y = Function("y")
ode = Eq(-a*x**2 - b*y(x)**2 + Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE -a*x**2 - b*y(x)**2 + Derivative(y(x), x) cannot be solved by the lie group method