2.22.45 Problem 53
Internal
problem
ID
[13540]
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
:
53
Date
solved
:
Friday, December 19, 2025 at 06:16:04 AM
CAS
classification
:
[_rational, [_Abel, `2nd type`, `class A`]]
\begin{align*}
y y^{\prime }-y&=-\frac {12 x}{49}+A \sqrt {x} \\
\end{align*}
Unknown ode type.
2.22.45.1 ✓ Maple. Time used: 0.002 (sec). Leaf size: 133
ode:=y(x)*diff(y(x),x)-y(x) = -12/49*x+A*x^(1/2);
dsolve(ode,y(x), singsol=all);
\[
\frac {\left (\left (\frac {4 \left (x -\frac {7 y}{4}\right ) \sqrt {3}\, \operatorname {hypergeom}\left (\left [\frac {1}{2}, \frac {7}{6}\right ], \left [\frac {3}{2}\right ], \frac {3 \left (-4 x +7 y\right )^{2}}{196 x^{{3}/{2}} A}\right )}{7}+c_1 \sqrt {x}\, \sqrt {A \sqrt {x}}\right ) 196^{{1}/{6}} \left (\frac {x^{{3}/{2}} A -\frac {12 \left (x -\frac {7 y}{4}\right )^{2}}{49}}{x^{{3}/{2}} A}\right )^{{1}/{6}}-7 \,14^{{1}/{3}} A \sqrt {3}\, \sqrt {x}\right ) 196^{{5}/{6}}}{196 \sqrt {x}\, \left (\frac {x^{{3}/{2}} A -\frac {12 \left (x -\frac {7 y}{4}\right )^{2}}{49}}{x^{{3}/{2}} A}\right )^{{1}/{6}} \sqrt {A \sqrt {x}}} = 0
\]
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
Looking for potential symmetries
<- Abel successful
Maple step by step
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y \left (x \right ) \left (\frac {d}{d x}y \left (x \right )\right )-y \left (x \right )=-\frac {12 x}{49}+A \sqrt {x} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=\frac {y \left (x \right )-\frac {12 x}{49}+A \sqrt {x}}{y \left (x \right )} \end {array} \]
2.22.45.2 ✗ Mathematica
ode=y[x]*D[y[x],x]-y[x]==-12/49*x+A*x^(1/2);
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
Not solved
2.22.45.3 ✗ Sympy
from sympy import *
x = symbols("x")
A = symbols("A")
y = Function("y")
ode = Eq(-A*sqrt(x) + 12*x/49 + y(x)*Derivative(y(x), x) - y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out