2.22.49 Problem 57
Internal
problem
ID
[13544]
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
:
57
Date
solved
:
Friday, December 19, 2025 at 06:20:21 AM
CAS
classification
:
[_rational, [_Abel, `2nd type`, `class B`]]
\begin{align*}
y y^{\prime }-y&=-\frac {10 x}{49}+\frac {2 A \left (4 \sqrt {x}+61 A +\frac {12 A^{2}}{\sqrt {x}}\right )}{49} \\
\end{align*}
Unknown ode type.
2.22.49.1 ✓ Maple. Time used: 0.002 (sec). Leaf size: 200
ode:=y(x)*diff(y(x),x)-y(x) = -10/49*x+2/49*A*(4*x^(1/2)+61*A+12*A^2/x^(1/2));
dsolve(ode,y(x), singsol=all);
\[
c_1 -\frac {\left (\sqrt {x}+3 A \right ) 2^{{2}/{3}} \left (\frac {3 A^{2}+16 A \sqrt {x}+5 x -7 y}{6 A^{2}+2 A \sqrt {x}+y}\right )^{{5}/{6}} y}{2 \sqrt {\frac {\left (\sqrt {x}+3 A \right )^{2}}{6 A^{2}+2 A \sqrt {x}+y}}\, \left (\frac {-24 A^{2}-2 A \sqrt {x}+2 x -7 y}{6 A^{2}+2 A \sqrt {x}+y}\right )^{{1}/{3}} \left (6 A^{2}+2 A \sqrt {x}+y\right ) A}-\int _{}^{\frac {6 A \sqrt {x}+2 x -3 y}{12 A^{2}+4 A \sqrt {x}+2 y}}\frac {\left (10 \textit {\_a} +1\right )^{{5}/{6}}}{\sqrt {2 \textit {\_a} +3}\, \left (\textit {\_a} -2\right )^{{1}/{3}}}d \textit {\_a} = 0
\]
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
Looking for potential symmetries
found: 2 potential symmetries. Proceeding with integration step
<- Abel successful
Maple step by step
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y \left (x \right ) \left (\frac {d}{d x}y \left (x \right )\right )-y \left (x \right )=-\frac {10 x}{49}+\frac {2 A \left (4 \sqrt {x}+61 A +\frac {12 A^{2}}{\sqrt {x}}\right )}{49} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=\frac {y \left (x \right )-\frac {10 x}{49}+\frac {2 A \left (4 \sqrt {x}+61 A +\frac {12 A^{2}}{\sqrt {x}}\right )}{49}}{y \left (x \right )} \end {array} \]
2.22.49.2 ✗ Mathematica
ode=y[x]*D[y[x],x]-y[x]==-10/49*x+2/49*A*(4*x^(1/2)+61*A+12*A^2*x^(-1/2));
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
Not solved
2.22.49.3 ✗ Sympy
from sympy import *
x = symbols("x")
A = symbols("A")
y = Function("y")
ode = Eq(-2*A*(12*A**2/sqrt(x) + 61*A + 4*sqrt(x))/49 + 10*x/49 + y(x)*Derivative(y(x), x) - y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out