55.24.17 problem 17

Internal problem ID [13646]
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 1.3.3-2.
Problem number : 17
Date solved : Wednesday, October 01, 2025 at 10:44:06 PM
CAS classification : [[_Abel, `2nd type`, `class B`]]

\begin{align*} y y^{\prime }&=\frac {3 y}{\left (a x +b \right )^{{1}/{3}} x^{{5}/{3}}}+\frac {3}{\left (a x +b \right )^{{2}/{3}} x^{{7}/{3}}} \end{align*}

✓ Maple. Time used: 0.009 (sec). Leaf size: 143

ode:=y(x)*diff(y(x),x) = 3/(a*x+b)^(1/3)/x^(5/3)*y(x)+3/(a*x+b)^(2/3)/x^(7/3); 
dsolve(ode,y(x), singsol=all);
 

\[ y = -\frac {6 \sqrt {3}}{\left (a x +b \right )^{{1}/{3}} x^{{2}/{3}} \left (\left (\left (a x +b \right )^{{1}/{3}} x^{{5}/{3}} \left (\frac {a}{\left (a x +b \right )^{2} x^{4}}\right )^{{1}/{3}}+2\right ) \sqrt {3}+3 \left (a x +b \right )^{{1}/{3}} x^{{5}/{3}} \left (\frac {a}{\left (a x +b \right )^{2} x^{4}}\right )^{{1}/{3}} \tan \left (\operatorname {RootOf}\left (\sqrt {3}\, \ln \left (\frac {\tan \left (\textit {\_Z} \right )^{2}+1}{\left (\sqrt {3}+\tan \left (\textit {\_Z} \right )\right )^{2}}\right )+6 \sqrt {3}\, c_1 -2 \sqrt {3}\, \int \left (\frac {a}{\left (a x +b \right )^{2} x^{4}}\right )^{{2}/{3}} \left (a x +b \right )^{{2}/{3}} x^{{7}/{3}}d x -6 \textit {\_Z} \right )\right )\right )} \]

✓ Mathematica. Time used: 1.221 (sec). Leaf size: 304

ode=y[x]*D[y[x],x]==3*(a*x+b)^(-1/3)*x^(-5/3)*y[x]+3*(a*x+b)^(-2/3)*x^(-7/3); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\frac {1}{6} \left (\frac {\sqrt [3]{a} x \left (\log \left (a^{2/3} x^{2/3}+\sqrt [3]{a} \sqrt [3]{x} \sqrt [3]{a x+b}+(a x+b)^{2/3}\right )+2 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{a x+b}}{\sqrt [3]{a x+b}+2 \sqrt [3]{a} \sqrt [3]{x}}\right )-2 \log \left (\sqrt [3]{a x+b}-\sqrt [3]{a} \sqrt [3]{x}\right )\right )}{\sqrt [3]{a x^3}}+2 \sqrt {3} \arctan \left (\frac {-\frac {2 \left (x^{2/3} y(x) \sqrt [3]{a x+b}+3\right )}{\sqrt [3]{a x^3} y(x)}-1}{\sqrt {3}}\right )+2 \log \left (\frac {-x^{2/3} y(x) \sqrt [3]{a x+b}-3}{\sqrt [3]{a x^3} y(x)}+1\right )-\log \left (\frac {\left (x^{2/3} y(x) \sqrt [3]{a x+b}+3\right )^2}{\left (a x^3\right )^{2/3} y(x)^2}+\frac {x^{2/3} y(x) \sqrt [3]{a x+b}+3}{\sqrt [3]{a x^3} y(x)}+1\right )\right )=c_1,y(x)\right ] \]

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
y = Function("y") 
ode = Eq(y(x)*Derivative(y(x), x) - 3*y(x)/(x**(5/3)*(a*x + b)**(1/3)) - 3/(x**(7/3)*(a*x + b)**(2/3)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out