2.22.42 Problem 49
Internal
problem
ID
[13537]
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
:
49
Date
solved
:
Friday, December 19, 2025 at 06:08:10 AM
CAS
classification
:
[_rational, [_Abel, `2nd type`, `class B`]]
\begin{align*}
y y^{\prime }-y&=2 x +2 A \left (10 \sqrt {x}+31 A +\frac {30 A^{2}}{\sqrt {x}}\right ) \\
\end{align*}
Unknown ode type.
2.22.42.1 ✓ Maple. Time used: 0.001 (sec). Leaf size: 196
ode:=y(x)*diff(y(x),x)-y(x) = 2*x+2*A*(10*x^(1/2)+31*A+30*A^2/x^(1/2));
dsolve(ode,y(x), singsol=all);
\[
c_1 -\frac {\left (\sqrt {x}+3 A \right ) 2^{{1}/{3}} \left (\frac {12 A^{2}+10 A \sqrt {x}+2 x -y}{6 A^{2}+2 A \sqrt {x}+y}\right )^{{1}/{3}} \left (\frac {15 A^{2}+8 A \sqrt {x}+x +y}{6 A^{2}+2 A \sqrt {x}+y}\right )^{{1}/{6}} y}{4 \sqrt {\frac {\left (\sqrt {x}+3 A \right )^{2}}{6 A^{2}+2 A \sqrt {x}+y}}\, \left (6 A^{2}+2 A \sqrt {x}+y\right ) A}-\int _{}^{\frac {6 A \sqrt {x}+2 x -3 y}{12 A^{2}+4 A \sqrt {x}+2 y}}\frac {\left (\textit {\_a} +1\right )^{{1}/{3}} \left (2 \textit {\_a} +5\right )^{{1}/{6}}}{\sqrt {2 \textit {\_a} +3}}d \textit {\_a} = 0
\]
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
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y \left (x \right ) \left (\frac {d}{d x}y \left (x \right )\right )-y \left (x \right )=2 x +2 A \left (10 \sqrt {x}+31 A +\frac {30 A^{2}}{\sqrt {x}}\right ) \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=\frac {y \left (x \right )+2 x +2 A \left (10 \sqrt {x}+31 A +\frac {30 A^{2}}{\sqrt {x}}\right )}{y \left (x \right )} \end {array} \]
2.22.42.2 ✗ Mathematica
ode=y[x]*D[y[x],x]-y[x]==2*x+2*A*(10*x^(1/2)+31*A+30*A^2*x^(-1/2));
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
Not solved
2.22.42.3 ✗ Sympy
from sympy import *
x = symbols("x")
A = symbols("A")
y = Function("y")
ode = Eq(-2*A*(30*A**2/sqrt(x) + 31*A + 10*sqrt(x)) - 2*x + y(x)*Derivative(y(x), x) - y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out