55.26.1 problem 1

Internal problem ID [13748]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 1, section 1.4. Equations Containing Polynomial Functions of y. subsection 1.4.1-2 Abel equations of the first kind.
Problem number : 1
Date solved : Thursday, October 02, 2025 at 07:57:33 AM
CAS classification : [[_homogeneous, `class G`], _rational, _Abel]

\begin{align*} y^{\prime }&=a y^{3}+\frac {b}{x^{{3}/{2}}} \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 34

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

\[ y = \frac {\operatorname {RootOf}\left (-\ln \left (x \right )+c_1 +2 \int _{}^{\textit {\_Z}}\frac {1}{2 a \,\textit {\_a}^{3}+\textit {\_a} +2 b}d \textit {\_a} \right )}{\sqrt {x}} \]

✓ Mathematica. Time used: 0.143 (sec). Leaf size: 320

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

\[ \text {Solve}\left [\frac {2}{3} a b^2 \text {RootSum}\left [8 \text {$\#$1}^9 a b^2+24 \text {$\#$1}^6 a b^2+24 \text {$\#$1}^3 a b^2+\text {$\#$1}^3+8 a b^2\&,\frac {4 \text {$\#$1}^6 \log \left (y(x) \sqrt [3]{\frac {a x^{3/2}}{b}}-\text {$\#$1}\right )+2 \text {$\#$1}^4 \sqrt [3]{-\frac {1}{a b^2}} \log \left (y(x) \sqrt [3]{\frac {a x^{3/2}}{b}}-\text {$\#$1}\right )+8 \text {$\#$1}^3 \log \left (y(x) \sqrt [3]{\frac {a x^{3/2}}{b}}-\text {$\#$1}\right )+\text {$\#$1}^2 \left (-\frac {1}{a b^2}\right )^{2/3} \log \left (y(x) \sqrt [3]{\frac {a x^{3/2}}{b}}-\text {$\#$1}\right )+2 \text {$\#$1} \sqrt [3]{-\frac {1}{a b^2}} \log \left (y(x) \sqrt [3]{\frac {a x^{3/2}}{b}}-\text {$\#$1}\right )+4 \log \left (y(x) \sqrt [3]{\frac {a x^{3/2}}{b}}-\text {$\#$1}\right )}{24 \text {$\#$1}^8 a b^2+48 \text {$\#$1}^5 a b^2+24 \text {$\#$1}^2 a b^2+\text {$\#$1}^2}\&\right ]=\frac {a x \log (x)}{\left (\frac {a x^{3/2}}{b}\right )^{2/3}}+c_1,y(x)\right ] \]

✗ Sympy

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

NotImplementedError : The given ODE -a*y(x)**3 - b/x**(3/2) + Derivative(y(x), x) cannot be solved b