55.23.4 problem 4

Internal problem ID [13621]
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, October 01, 2025 at 07:40:26 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: 0.191 (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);
 

\[ \frac {2 \,\operatorname {arctanh}\left (\frac {-\sqrt {a x +b}\, y a +a x +b}{\sqrt {\left (2 a +1\right ) \left (a x +b \right )^{2}}}\right ) a x}{\sqrt {\left (2 a +1\right ) \left (a x +b \right )^{2}}}+\frac {2 \,\operatorname {arctanh}\left (\frac {-\sqrt {a x +b}\, y a +a x +b}{\sqrt {\left (2 a +1\right ) \left (a x +b \right )^{2}}}\right ) b}{\sqrt {\left (2 a +1\right ) \left (a x +b \right )^{2}}}+\ln \left (\left (a y^{2}-2 a x -2 b \right ) \sqrt {a x +b}-2 y \left (a x +b \right )\right )-\frac {\ln \left (a x +b \right )}{2}-c_1 = 0 \]

✓ Mathematica. Time used: 0.091 (sec). Leaf size: 90

ode=y[x]*D[y[x],x]==(a*x+b)^(-1/2)*y[x]+1; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [-\frac {\frac {2 \arctan \left (\frac {\frac {a y(x)}{\sqrt {a x+b}}-1}{\sqrt {-2 a-1}}\right )}{\sqrt {-2 a-1}}+\log \left (-\frac {a y(x)^2}{a x+b}+\frac {2 y(x)}{\sqrt {a x+b}}+2\right )}{a}=\frac {\log (a x+b)}{a}+c_1,y(x)\right ] \]

✓ Sympy. Time used: 4.426 (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)
 

\[ C_{1} + \frac {\left (1 - \frac {1}{\sqrt {2 a + 1}}\right ) \log {\left (\frac {y{\left (x \right )}}{\sqrt {a x + b}} + \frac {2}{\sqrt {2 a + 1}} - \frac {1}{a} + \frac {1}{a \sqrt {2 a + 1}} \right )}}{2} + \frac {\left (1 + \frac {1}{\sqrt {2 a + 1}}\right ) \log {\left (\frac {y{\left (x \right )}}{\sqrt {a x + b}} - \frac {2}{\sqrt {2 a + 1}} - \frac {1}{a} - \frac {1}{a \sqrt {2 a + 1}} \right )}}{2} + \frac {\log {\left (2 a x + 2 b \right )}}{2} = 0 \]