23.1.281 problem 275
Internal
problem
ID
[4888]
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
:
275
Date
solved
:
Sunday, October 12, 2025 at 01:18:22 AM
CAS
classification
:
[_rational, _Abel]
\begin{align*} x^{2} y^{\prime }&=a \,x^{2} y^{2}-a y^{3} \end{align*}
✓ Maple. Time used: 0.002 (sec). Leaf size: 148
ode:=x^2*diff(y(x),x) = a*x^2*y(x)^2-a*y(x)^3;
dsolve(ode,y(x), singsol=all);
\[
y = -\frac {1}{a x +2^{{2}/{3}} \left (-a \right )^{{2}/{3}} \operatorname {RootOf}\left (\operatorname {AiryBi}\left (\frac {\left (\textit {\_Z}^{2} 2^{{1}/{3}} \left (-a \right )^{{1}/{3}} x -1\right ) 2^{{2}/{3}}}{2 \left (-a \right )^{{1}/{3}} x}\right ) c_1 \textit {\_Z} +\textit {\_Z} \operatorname {AiryAi}\left (\frac {\left (\textit {\_Z}^{2} 2^{{1}/{3}} \left (-a \right )^{{1}/{3}} x -1\right ) 2^{{2}/{3}}}{2 \left (-a \right )^{{1}/{3}} x}\right )+\operatorname {AiryBi}\left (1, \frac {\left (\textit {\_Z}^{2} 2^{{1}/{3}} \left (-a \right )^{{1}/{3}} x -1\right ) 2^{{2}/{3}}}{2 \left (-a \right )^{{1}/{3}} x}\right ) c_1 +\operatorname {AiryAi}\left (1, \frac {\left (\textit {\_Z}^{2} 2^{{1}/{3}} \left (-a \right )^{{1}/{3}} x -1\right ) 2^{{2}/{3}}}{2 \left (-a \right )^{{1}/{3}} x}\right )\right )}
\]
✓ Mathematica. Time used: 8.62 (sec). Leaf size: 303
ode=x^2*D[y[x],x]==a*x^2*y[x]^2-a*y[x]^3;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
\text {Solve}\left [\frac {\left (-\frac {\left (-\frac {1}{2}\right )^{2/3}}{a^{2/3} y(x)}-\left (-\frac {1}{2}\right )^{2/3} \sqrt [3]{a} x\right ) \operatorname {AiryAi}\left (\left (-\left (-\frac {1}{2}\right )^{2/3} \sqrt [3]{a} x-\frac {\left (-\frac {1}{2}\right )^{2/3}}{a^{2/3} y(x)}\right )^2-\frac {\sqrt [3]{-\frac {1}{2}}}{\sqrt [3]{a} x}\right )+\operatorname {AiryAiPrime}\left (\left (-\left (-\frac {1}{2}\right )^{2/3} \sqrt [3]{a} x-\frac {\left (-\frac {1}{2}\right )^{2/3}}{a^{2/3} y(x)}\right )^2-\frac {\sqrt [3]{-\frac {1}{2}}}{\sqrt [3]{a} x}\right )}{\left (-\frac {\left (-\frac {1}{2}\right )^{2/3}}{a^{2/3} y(x)}-\left (-\frac {1}{2}\right )^{2/3} \sqrt [3]{a} x\right ) \operatorname {AiryBi}\left (\left (-\left (-\frac {1}{2}\right )^{2/3} \sqrt [3]{a} x-\frac {\left (-\frac {1}{2}\right )^{2/3}}{a^{2/3} y(x)}\right )^2-\frac {\sqrt [3]{-\frac {1}{2}}}{\sqrt [3]{a} x}\right )+\operatorname {AiryBiPrime}\left (\left (-\left (-\frac {1}{2}\right )^{2/3} \sqrt [3]{a} x-\frac {\left (-\frac {1}{2}\right )^{2/3}}{a^{2/3} y(x)}\right )^2-\frac {\sqrt [3]{-\frac {1}{2}}}{\sqrt [3]{a} x}\right )}+c_1=0,y(x)\right ]
\]
✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
y = Function("y")
ode = Eq(-a*x**2*y(x)**2 + a*y(x)**3 + x**2*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
RecursionError : maximum recursion depth exceeded