2.22.39 Problem 46
Internal
problem
ID
[13534]
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
:
46
Date
solved
:
Friday, December 19, 2025 at 06:04:03 AM
CAS
classification
:
[_rational, [_Abel, `2nd type`, `class A`]]
\begin{align*}
y y^{\prime }-y&=\frac {6}{25} x -A \,x^{2} \\
\end{align*}
Unknown ode type.
2.22.39.1 ✓ Maple. Time used: 0.002 (sec). Leaf size: 180
ode:=y(x)*diff(y(x),x)-y(x) = 6/25*x-A*x^2;
dsolve(ode,y(x), singsol=all);
\[
-\frac {125 \left (5^{{1}/{3}} 2^{{5}/{6}} {\left (-\frac {1250 \left (\frac {3 A y^{2}}{2}+\left (-\frac {6 A x}{5}+\frac {36}{125}\right ) y+\left (A x -\frac {6}{25}\right )^{2} x \right ) A}{\left (50 A x -125 A y-12\right )^{2}}\right )}^{{1}/{6}} A y \sqrt {-25 A x +6}-\frac {4 \left (\int _{}^{\frac {2 \left (-25 A x +6\right )^{{3}/{2}}}{-50 A x +125 A y+12}}\frac {\left (\textit {\_a}^{2}-6\right )^{{1}/{6}}}{\textit {\_a}^{{1}/{3}}}d \textit {\_a} +c_1 \right ) \left (A x -\frac {5 A y}{2}-\frac {6}{25}\right ) \left (\frac {\left (-25 A x +6\right )^{{3}/{2}}}{-50 A x +125 A y+12}\right )^{{1}/{3}}}{5}\right )}{\left (\frac {\left (-25 A x +6\right )^{{3}/{2}}}{-50 A x +125 A y+12}\right )^{{1}/{3}} \left (100 A x -250 A y-24\right )} = 0
\]
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
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y \left (x \right ) \left (\frac {d}{d x}y \left (x \right )\right )-y \left (x \right )=\frac {6}{25} x -A \,x^{2} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=\frac {y \left (x \right )+\frac {6 x}{25}-A \,x^{2}}{y \left (x \right )} \end {array} \]
2.22.39.2 ✓ Mathematica. Time used: 1.309 (sec). Leaf size: 189
ode=y[x]*D[y[x],x]-y[x]==6/25*x-A*x^2;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
\text {Solve}\left [\frac {\sqrt [3]{5} \sqrt [6]{-\frac {A \left (1875 A y(x)^2-60 (25 A x-6) y(x)+2 x (6-25 A x)^2\right )}{(25 A x-6)^3}} \left (\frac {(-125 A y(x)+50 A x-12) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {5}{6},\frac {3}{2},-\frac {3 (-50 A x+125 A y(x)+12)^2}{2 (25 A x-6)^3}\right )}{\sqrt [3]{10} \sqrt {18-75 A x} (25 A x-6) \sqrt [6]{\frac {A \left (1875 A y(x)^2-60 (25 A x-6) y(x)+2 x (6-25 A x)^2\right )}{(25 A x-6)^3}}}+\sqrt {1-\frac {25 A x}{6}}\right )}{\sqrt [6]{2}}+c_1=0,y(x)\right ]
\]
2.22.39.3 ✗ Sympy
from sympy import *
x = symbols("x")
A = symbols("A")
y = Function("y")
ode = Eq(A*x**2 - 6*x/25 + y(x)*Derivative(y(x), x) - y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out
Python version: 3.12.3 (main, Aug 14 2025, 17:47:21) [GCC 13.3.0]
Sympy version 1.14.0