23.1.413 problem 402
Internal
problem
ID
[5020]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Part
II.
Chapter
1.
THE
DIFFERENTIAL
EQUATION
IS
OF
FIRST
ORDER
AND
OF
FIRST
DEGREE,
page
223
Problem
number
:
402
Date
solved
:
Tuesday, September 30, 2025 at 11:23:32 AM
CAS
classification
:
[_rational, [_Riccati, _special]]
\begin{align*} x^{{3}/{2}} y^{\prime }&=a +b \,x^{{3}/{2}} y^{2} \end{align*}
✓ Maple. Time used: 0.004 (sec). Leaf size: 122
ode:=x^(3/2)*diff(y(x),x) = a+b*x^(3/2)*y(x)^2;
dsolve(ode,y(x), singsol=all);
\[
y = \frac {2 a \left (\operatorname {BesselJ}\left (1, 4 \sqrt {a}\, \sqrt {b}\, x^{{1}/{4}}\right ) c_1 +\operatorname {BesselY}\left (1, 4 \sqrt {a}\, \sqrt {b}\, x^{{1}/{4}}\right )\right )}{\sqrt {x}\, \left (2 \operatorname {BesselJ}\left (0, 4 \sqrt {a}\, \sqrt {b}\, x^{{1}/{4}}\right ) \sqrt {a}\, \sqrt {b}\, x^{{1}/{4}} c_1 +2 \operatorname {BesselY}\left (0, 4 \sqrt {a}\, \sqrt {b}\, x^{{1}/{4}}\right ) \sqrt {a}\, \sqrt {b}\, x^{{1}/{4}}-\operatorname {BesselJ}\left (1, 4 \sqrt {a}\, \sqrt {b}\, x^{{1}/{4}}\right ) c_1 -\operatorname {BesselY}\left (1, 4 \sqrt {a}\, \sqrt {b}\, x^{{1}/{4}}\right )\right )}
\]
✓ Mathematica. Time used: 0.165 (sec). Leaf size: 373
ode=x^(3/2)*D[y[x],x]==a+ b*x^(3/2)*y[x]^2;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to -\frac {\sqrt {a} \sqrt {b} \sqrt [4]{x} \operatorname {BesselY}\left (1,4 \sqrt {a} \sqrt {b} \sqrt [4]{x}\right )+\operatorname {BesselY}\left (2,4 \sqrt {a} \sqrt {b} \sqrt [4]{x}\right )-\sqrt {a} \sqrt {b} \sqrt [4]{x} \operatorname {BesselY}\left (3,4 \sqrt {a} \sqrt {b} \sqrt [4]{x}\right )-\sqrt {a} \sqrt {b} c_1 \sqrt [4]{x} \operatorname {BesselJ}\left (1,4 \sqrt {a} \sqrt {b} \sqrt [4]{x}\right )-c_1 \operatorname {BesselJ}\left (2,4 \sqrt {a} \sqrt {b} \sqrt [4]{x}\right )+\sqrt {a} \sqrt {b} c_1 \sqrt [4]{x} \operatorname {BesselJ}\left (3,4 \sqrt {a} \sqrt {b} \sqrt [4]{x}\right )}{2 b x \operatorname {BesselY}\left (2,4 \sqrt {a} \sqrt {b} \sqrt [4]{x}\right )-2 b c_1 x \operatorname {BesselJ}\left (2,4 \sqrt {a} \sqrt {b} \sqrt [4]{x}\right )}\\ y(x)&\to -\frac {\sqrt {a} \sqrt {b} \sqrt [4]{x} \operatorname {BesselJ}\left (1,4 \sqrt {a} \sqrt {b} \sqrt [4]{x}\right )+\operatorname {BesselJ}\left (2,4 \sqrt {a} \sqrt {b} \sqrt [4]{x}\right )-\sqrt {a} \sqrt {b} \sqrt [4]{x} \operatorname {BesselJ}\left (3,4 \sqrt {a} \sqrt {b} \sqrt [4]{x}\right )}{2 b x \operatorname {BesselJ}\left (2,4 \sqrt {a} \sqrt {b} \sqrt [4]{x}\right )} \end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
b = symbols("b")
y = Function("y")
ode = Eq(-a - b*x**(3/2)*y(x)**2 + x**(3/2)*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE -a/x**(3/2) - b*y(x)**2 + Derivative(y(x), x) cannot be solved by the factorable group method