55.22.33 problem 33
Internal
problem
ID
[13574]
Book
:
Handbook
of
exact
solutions
for
ordinary
differential
equations.
By
Polyanin
and
Zaitsev.
Second
edition
Section
:
Chapter
1,
section
1.3.
Abel
Equations
of
the
Second
Kind.
subsection
1.3.1-2.
Solvable
equations
and
their
solutions
Problem
number
:
33
Date
solved
:
Wednesday, October 01, 2025 at 05:49:51 PM
CAS
classification
:
[_rational, [_Abel, `2nd type`, `class B`]]
\begin{align*} y y^{\prime }-y&=\frac {A}{x^{2}} \end{align*}
✓ Maple. Time used: 0.002 (sec). Leaf size: 276
ode:=y(x)*diff(y(x),x)-y(x) = A/x^2;
dsolve(ode,y(x), singsol=all);
\[
\frac {\left (x -y\right ) A \left (\operatorname {AiryBi}\left (-\frac {\left (x^{3}-2 y x^{2}+x y^{2}+2 A \right ) 2^{{2}/{3}}}{4 \left (-A^{2}\right )^{{1}/{3}} x}\right ) c_1 -\operatorname {AiryAi}\left (-\frac {\left (x^{3}-2 y x^{2}+x y^{2}+2 A \right ) 2^{{2}/{3}}}{4 \left (-A^{2}\right )^{{1}/{3}} x}\right )\right ) 2^{{1}/{3}}+2 \left (-A^{2}\right )^{{2}/{3}} \left (-\operatorname {AiryBi}\left (1, -\frac {\left (x^{3}-2 y x^{2}+x y^{2}+2 A \right ) 2^{{2}/{3}}}{4 \left (-A^{2}\right )^{{1}/{3}} x}\right ) c_1 +\operatorname {AiryAi}\left (1, -\frac {\left (x^{3}-2 y x^{2}+x y^{2}+2 A \right ) 2^{{2}/{3}}}{4 \left (-A^{2}\right )^{{1}/{3}} x}\right )\right )}{A 2^{{1}/{3}} \left (x -y\right ) \operatorname {AiryBi}\left (-\frac {\left (x^{3}-2 y x^{2}+x y^{2}+2 A \right ) 2^{{2}/{3}}}{4 \left (-A^{2}\right )^{{1}/{3}} x}\right )-2 \operatorname {AiryBi}\left (1, -\frac {\left (x^{3}-2 y x^{2}+x y^{2}+2 A \right ) 2^{{2}/{3}}}{4 \left (-A^{2}\right )^{{1}/{3}} x}\right ) \left (-A^{2}\right )^{{2}/{3}}} = 0
\]
✓ Mathematica. Time used: 2.595 (sec). Leaf size: 213
ode=y[x]*D[y[x],x]-y[x]==A*x^(-2);
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
\text {Solve}\left [\frac {\operatorname {AiryAiPrime}\left (-\frac {\sqrt [3]{-\frac {1}{2}} \left (x^3-2 y(x) x^2+y(x)^2 x+2 A\right )}{2 A^{2/3} x}\right )-\frac {\left (-\frac {1}{2}\right )^{2/3} (x-y(x)) \operatorname {AiryAi}\left (-\frac {\sqrt [3]{-\frac {1}{2}} \left (x^3-2 y(x) x^2+y(x)^2 x+2 A\right )}{2 A^{2/3} x}\right )}{\sqrt [3]{A}}}{\operatorname {AiryBiPrime}\left (-\frac {\sqrt [3]{-\frac {1}{2}} \left (x^3-2 y(x) x^2+y(x)^2 x+2 A\right )}{2 A^{2/3} x}\right )-\frac {\left (-\frac {1}{2}\right )^{2/3} (x-y(x)) \operatorname {AiryBi}\left (-\frac {\sqrt [3]{-\frac {1}{2}} \left (x^3-2 y(x) x^2+y(x)^2 x+2 A\right )}{2 A^{2/3} x}\right )}{\sqrt [3]{A}}}+c_1=0,y(x)\right ]
\]
✗ Sympy
from sympy import *
x = symbols("x")
A = symbols("A")
y = Function("y")
ode = Eq(-A/x**2 + y(x)*Derivative(y(x), x) - y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE -A/(x**2*y(x)) + Derivative(y(x), x) - 1 cannot be solved by the factorable group method