2.22.38 Problem 45
Internal
problem
ID
[13533]
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
:
45
Date
solved
:
Friday, December 19, 2025 at 06:02:56 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.
2.22.38.1 ✓ Maple. Time used: 0.001 (sec). Leaf size: 132
ode:=y(x)*diff(y(x),x)-y(x) = -6/25*x-A*x^2;
dsolve(ode,y(x), singsol=all);
\[
c_1 +\frac {\left (2 x -5 y\right ) \int _{}^{-\frac {10 \sqrt {-A x}\, x}{-2 x +5 y}}\frac {\left (\textit {\_a}^{2}-6\right )^{{1}/{6}}}{\textit {\_a}^{{1}/{3}}}d \textit {\_a} -\frac {5 \,2^{{5}/{6}} 5^{{2}/{3}} \left (\frac {-50 A \,x^{3}-12 x^{2}+60 x y-75 y^{2}}{\left (-2 x +5 y\right )^{2}}\right )^{{1}/{6}} \sqrt {-A x}\, y}{2 \left (-\frac {\sqrt {-A x}\, x}{-2 x +5 y}\right )^{{1}/{3}}}}{2 x -5 y} = 0
\]
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
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y \left (x \right ) \left (\frac {d}{d x}y \left (x \right )\right )-y \left (x \right )=-\frac {6}{25} x -A \,x^{2} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=\frac {y \left (x \right )-\frac {6 x}{25}-A \,x^{2}}{y \left (x \right )} \end {array} \]
2.22.38.2 ✓ Mathematica. Time used: 0.838 (sec). Leaf size: 162
ode=y[x]*D[y[x],x]-y[x]==-6/25*x-A*x^2;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
\text {Solve}\left [c_1=\frac {i \sqrt [6]{\frac {-2 x^2 (25 A x+6)+60 x y(x)-75 y(x)^2}{A x^3}} \left (25 A x^2-\frac {\sqrt [6]{2} \sqrt [3]{5} (2 x-5 y(x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {5}{6},\frac {3}{2},-\frac {3 (2 x-5 y(x))^2}{50 A x^3}\right )}{\sqrt [6]{\frac {2 x^2 (25 A x+6)-60 x y(x)+75 y(x)^2}{A x^3}}}\right )}{5\ 2^{2/3} \sqrt {3} \sqrt [3]{5} \sqrt {A} x^{3/2}},y(x)\right ]
\]
2.22.38.3 ✗ 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