61.23.4 problem 4
Internal
problem
ID
[12326]
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
1.3.2.
Problem
number
:
4
Date
solved
:
Wednesday, March 05, 2025 at 06:41:41 PM
CAS
classification
:
[[_1st_order, _with_linear_symmetries], [_Abel, `2nd type`, `class B`]]
\begin{align*} y y^{\prime }&=\frac {y}{\sqrt {a x +b}}+1 \end{align*}
✓ Maple. Time used: 0.437 (sec). Leaf size: 153
ode:=y(x)*diff(y(x),x) = 1/(a*x+b)^(1/2)*y(x)+1;
dsolve(ode,y(x), singsol=all);
\[
\frac {2 \,\operatorname {arctanh}\left (\frac {-\sqrt {a x +b}\, y a +a x +b}{\sqrt {\left (2 a +1\right ) \left (a x +b \right )^{2}}}\right ) a x}{\sqrt {\left (2 a +1\right ) \left (a x +b \right )^{2}}}+\ln \left (\left (a y^{2}-2 a x -2 b \right ) \sqrt {a x +b}-2 \left (a x +b \right ) y\right )+\frac {2 \,\operatorname {arctanh}\left (\frac {-\sqrt {a x +b}\, y a +a x +b}{\sqrt {\left (2 a +1\right ) \left (a x +b \right )^{2}}}\right ) b}{\sqrt {\left (2 a +1\right ) \left (a x +b \right )^{2}}}-\frac {\ln \left (a x +b \right )}{2}-c_{1} = 0
\]
✓ Mathematica. Time used: 0.072 (sec). Leaf size: 50
ode=y[x]*D[y[x],x]==(a*x+b)^(-1/2)*y[x]+1;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
\text {Solve}\left [\int _1^{\frac {y(x)}{\sqrt {b+a x}}}\frac {1}{-\frac {1}{2} a K[1]+1+\frac {1}{K[1]}}dK[1]=\frac {\log (a x+b)}{a}+c_1,y(x)\right ]
\]
✓ Sympy. Time used: 7.639 (sec). Leaf size: 119
from sympy import *
x = symbols("x")
a = symbols("a")
b = symbols("b")
y = Function("y")
ode = Eq(y(x)*Derivative(y(x), x) - 1 - y(x)/sqrt(a*x + b),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
C_{1} + \frac {\left (1 - \frac {1}{\sqrt {2 a + 1}}\right ) \log {\left (\frac {y{\left (x \right )}}{\sqrt {a x + b}} + \frac {2}{\sqrt {2 a + 1}} - \frac {1}{a} + \frac {1}{a \sqrt {2 a + 1}} \right )}}{2} + \frac {\left (1 + \frac {1}{\sqrt {2 a + 1}}\right ) \log {\left (\frac {y{\left (x \right )}}{\sqrt {a x + b}} - \frac {2}{\sqrt {2 a + 1}} - \frac {1}{a} - \frac {1}{a \sqrt {2 a + 1}} \right )}}{2} + \frac {\log {\left (2 a x + 2 b \right )}}{2} = 0
\]