2.22.20 Problem 24
Internal
problem
ID
[13515]
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
:
24
Date
solved
:
Friday, December 19, 2025 at 05:25:04 AM
CAS
classification
:
[_rational, [_Abel, `2nd type`, `class B`]]
\begin{align*}
y y^{\prime }-y&=-\frac {12 x}{49}+\frac {2 A \left (5 \sqrt {x}+34 A +\frac {15 A^{2}}{\sqrt {x}}\right )}{49} \\
\end{align*}
Unknown ode type.
2.22.20.1 ✓ Maple. Time used: 0.002 (sec). Leaf size: 270
ode:=y(x)*diff(y(x),x)-y(x) = -12/49*x+2/49*A*(5*x^(1/2)+34*A+15*A^2/x^(1/2));
dsolve(ode,y(x), singsol=all);
\[
\frac {\left (-\sqrt {x}+3 A \right ) \left (36 A^{4}+120 A^{3} \sqrt {x}-80 A \,x^{{3}/{2}}+52 A^{2} x +84 A^{2} y+140 A \sqrt {x}\, y+16 x^{2}-56 x y+49 y^{2}\right ) y}{8 \left (\frac {15 A^{2}+4 \sqrt {x}\, A -3 x +7 y}{6 A^{2}-2 \sqrt {x}\, A +y}\right )^{{3}/{2}} \sqrt {-\frac {\left (-\sqrt {x}+3 A \right )^{2}}{6 A^{2}-2 \sqrt {x}\, A +y}}\, \left (6 A^{2}-2 \sqrt {x}\, A +y\right )^{3} A}+\frac {\left (-54 A^{2}-6 \sqrt {x}\, A +8 x -21 y\right ) \sqrt {-\frac {\left (-\sqrt {x}+3 A \right )^{2}}{6 A^{2}-2 \sqrt {x}\, A +y}}}{\sqrt {\frac {15 A^{2}+4 \sqrt {x}\, A -3 x +7 y}{6 A^{2}-2 \sqrt {x}\, A +y}}\, \left (36 A^{2}-12 \sqrt {x}\, A +6 y\right )}+c_1 = 0
\]
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
Looking for potential symmetries
found: 2 potential symmetries. Proceeding with integration step
<- Abel successful
Maple step by step
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y \left (x \right ) \left (\frac {d}{d x}y \left (x \right )\right )-y \left (x \right )=-\frac {12 x}{49}+\frac {2 A \left (5 \sqrt {x}+34 A +\frac {15 A^{2}}{\sqrt {x}}\right )}{49} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=\frac {y \left (x \right )-\frac {12 x}{49}+\frac {2 A \left (5 \sqrt {x}+34 A +\frac {15 A^{2}}{\sqrt {x}}\right )}{49}}{y \left (x \right )} \end {array} \]
2.22.20.2 ✗ Mathematica
ode=y[x]*D[y[x],x]-y[x]==-12/49*x+2/49*A*(5*x^(1/2)+34*A+15*A^2*x^(-1/2));
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
Not solved
2.22.20.3 ✗ Sympy
from sympy import *
x = symbols("x")
A = symbols("A")
y = Function("y")
ode = Eq(-2*A*(15*A**2/sqrt(x) + 34*A + 5*sqrt(x))/49 + 12*x/49 + 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