2.22.27 Problem 32
Internal
problem
ID
[13522]
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
:
32
Date
solved
:
Friday, December 19, 2025 at 05:39:07 AM
CAS
classification
:
[_rational, [_Abel, `2nd type`, `class B`]]
\begin{align*}
y y^{\prime }-y&=\frac {A}{\sqrt {x}} \\
\end{align*}
Unknown ode type.
2.22.27.1 ✓ Maple. Time used: 0.002 (sec). Leaf size: 222
ode:=y(x)*diff(y(x),x)-y(x) = A/x^(1/2);
dsolve(ode,y(x), singsol=all);
\[
\frac {2^{{2}/{3}} \left (\operatorname {AiryBi}\left (-\frac {2^{{1}/{3}} \left (-A^{2} x^{{3}/{2}}\right )^{{2}/{3}} \left (-x +y\right )}{2 A^{2} x}\right ) c_1 -\operatorname {AiryAi}\left (-\frac {2^{{1}/{3}} \left (-A^{2} x^{{3}/{2}}\right )^{{2}/{3}} \left (-x +y\right )}{2 A^{2} x}\right )\right ) \left (-A^{2} x^{{3}/{2}}\right )^{{1}/{3}}-2 \left (-\operatorname {AiryBi}\left (1, -\frac {2^{{1}/{3}} \left (-A^{2} x^{{3}/{2}}\right )^{{2}/{3}} \left (-x +y\right )}{2 A^{2} x}\right ) c_1 +\operatorname {AiryAi}\left (1, -\frac {2^{{1}/{3}} \left (-A^{2} x^{{3}/{2}}\right )^{{2}/{3}} \left (-x +y\right )}{2 A^{2} x}\right )\right ) A}{2^{{2}/{3}} \left (-A^{2} x^{{3}/{2}}\right )^{{1}/{3}} \operatorname {AiryBi}\left (-\frac {2^{{1}/{3}} \left (-A^{2} x^{{3}/{2}}\right )^{{2}/{3}} \left (-x +y\right )}{2 A^{2} x}\right )+2 \operatorname {AiryBi}\left (1, -\frac {2^{{1}/{3}} \left (-A^{2} x^{{3}/{2}}\right )^{{2}/{3}} \left (-x +y\right )}{2 A^{2} x}\right ) A} = 0
\]
Maple trace
Methods for first order ODEs:
--- Trying classification methods ---
trying a quadrature
trying 1st order linear
trying Bernoulli
trying separable
trying inverse linear
trying homogeneous types:
trying Chini
differential order: 1; looking for linear symmetries
trying exact
trying Abel
<- Abel successful
Maple step by step
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y \left (x \right ) \left (\frac {d}{d x}y \left (x \right )\right )-y \left (x \right )=\frac {A}{\sqrt {x}} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=\frac {y \left (x \right )+\frac {A}{\sqrt {x}}}{y \left (x \right )} \end {array} \]
2.22.27.2 ✓ Mathematica. Time used: 0.179 (sec). Leaf size: 139
ode=y[x]*D[y[x],x]-y[x]==A*x^(-1/2);
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
\text {Solve}\left [\frac {\sqrt [3]{-1} 2^{2/3} \sqrt {x} \operatorname {AiryAi}\left (\frac {\left (-\frac {1}{2}\right )^{2/3} (x-y(x))}{A^{2/3}}\right )+2 \sqrt [3]{A} \operatorname {AiryAiPrime}\left (\frac {\left (-\frac {1}{2}\right )^{2/3} (x-y(x))}{A^{2/3}}\right )}{\sqrt [3]{-1} 2^{2/3} \sqrt {x} \operatorname {AiryBi}\left (\frac {\left (-\frac {1}{2}\right )^{2/3} (x-y(x))}{A^{2/3}}\right )+2 \sqrt [3]{A} \operatorname {AiryBiPrime}\left (\frac {\left (-\frac {1}{2}\right )^{2/3} (x-y(x))}{A^{2/3}}\right )}+c_1=0,y(x)\right ]
\]
2.22.27.3 ✗ Sympy
from sympy import *
x = symbols("x")
A = symbols("A")
y = Function("y")
ode = Eq(-A/sqrt(x) + y(x)*Derivative(y(x), x) - y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE -A/(sqrt(x)*y(x)) + Derivative(y(x), x) - 1 cannot be solved by
Python version: 3.12.3 (main, Aug 14 2025, 17:47:21) [GCC 13.3.0]
Sympy version 1.14.0
classify_ode(ode,func=y(x))
('factorable', 'lie_group')