2.22.37 Problem 44
Internal
problem
ID
[13532]
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
:
44
Date
solved
:
Friday, December 19, 2025 at 06:02:06 AM
CAS
classification
:
[_rational, [_Abel, `2nd type`, `class A`]]
\begin{align*}
y y^{\prime }-y&=A \,x^{2}-\frac {9}{625 A} \\
\end{align*}
Unknown ode type.
2.22.37.1 ✓ Maple. Time used: 0.001 (sec). Leaf size: 186
ode:=y(x)*diff(y(x),x)-y(x) = A*x^2-9/625/A;
dsolve(ode,y(x), singsol=all);
\[
\frac {-\frac {125 \,2^{{5}/{6}} \left (\frac {-46875 A^{2} y^{2}+\left (37500 A^{2} x +4500 A \right ) y+31250 \left (A x -\frac {3}{25}\right ) \left (A x +\frac {3}{25}\right )^{2}}{\left (50 A x -125 y A +6\right )^{2}}\right )^{{1}/{6}} y A \sqrt {25 A x +3}}{2}+50 \left (\frac {\left (25 A x +3\right )^{{3}/{2}}}{50 A x -125 y A +6}\right )^{{1}/{3}} \left (\int _{}^{-\frac {2 \left (25 A x +3\right )^{{3}/{2}}}{125 y A -50 A x -6}}\frac {\left (\textit {\_a}^{2}-6\right )^{{1}/{6}}}{\textit {\_a}^{{1}/{3}}}d \textit {\_a} +c_1 \right ) \left (A x -\frac {5 y A}{2}+\frac {3}{25}\right )}{\left (\frac {\left (25 A x +3\right )^{{3}/{2}}}{50 A x -125 y A +6}\right )^{{1}/{3}} \left (50 A x -125 y A +6\right )} = 0
\]
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
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y \left (x \right ) \left (\frac {d}{d x}y \left (x \right )\right )-y \left (x \right )=A \,x^{2}-\frac {9}{625 A} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=\frac {y \left (x \right )+A \,x^{2}-\frac {9}{625 A}}{y \left (x \right )} \end {array} \]
2.22.37.2 ✓ Mathematica. Time used: 0.984 (sec). Leaf size: 198
ode=y[x]*D[y[x],x]-y[x]==A*x^2-9/625*A^(-1);
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
\text {Solve}\left [\frac {\sqrt [6]{\frac {46875 A^2 y(x)^2-1500 A (25 A x+3) y(x)-2 (25 A x-3) (25 A x+3)^2}{(25 A x+3)^3}} \left (\frac {(-125 A y(x)+50 A x+6) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {5}{6},\frac {3}{2},\frac {3 (50 A x-125 A y(x)+6)^2}{2 (25 A x+3)^3}\right )}{\sqrt [3]{2} \sqrt {3} (25 A x+3)^{3/2} \sqrt [6]{\frac {-46875 A^2 y(x)^2+1500 A (25 A x+3) y(x)+2 (25 A x-3) (25 A x+3)^2}{(25 A x+3)^3}}}+\frac {\sqrt {25 A x+3}}{\sqrt {6}}\right )}{\sqrt [6]{2}}+c_1=0,y(x)\right ]
\]
2.22.37.3 ✗ 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) + 9/(625*A),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out