55.24.31 problem 31

Internal problem ID [13660]
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 : 31
Date solved : Sunday, October 12, 2025 at 04:22:25 AM
CAS classification : [_rational, [_Abel, `2nd type`, `class B`]]

\begin{align*} y y^{\prime }-\frac {a \left (x +1\right ) y}{2 x^{{7}/{4}}}&=\frac {a^{2} \left (x -1\right ) \left (3 x +5\right )}{4 x^{{5}/{2}}} \end{align*}

✓ Maple. Time used: 0.005 (sec). Leaf size: 187

ode:=y(x)*diff(y(x),x)-1/2*a*(1+x)/x^(7/4)*y(x) = 1/4*a^2*(x-1)*(3*x+5)/x^(5/2); 
dsolve(ode,y(x), singsol=all);
 

\[ \frac {\frac {36 \sqrt {13}\, \sqrt {-\frac {\left (x -1\right ) a +x^{{3}/{4}} y}{x^{{3}/{4}} \left (y+a \,x^{{1}/{4}}\right )}}\, 55^{{1}/{6}} \left (x -\frac {15}{2}\right ) \left (\frac {\left (3 x +5\right ) a +3 x^{{3}/{4}} y}{x^{{3}/{4}} \left (y+a \,x^{{1}/{4}}\right )}\right )^{{5}/{6}}}{20449}+1458000 \left (\frac {a}{x^{{3}/{4}} \left (y+a \,x^{{1}/{4}}\right )}\right )^{{4}/{3}} \left (\int _{}^{-\frac {90 \left (2 x^{{3}/{4}} y+2 a x -15 a \right )}{143 \left (x^{{3}/{4}} y+a x \right )}}\frac {\textit {\_a} \left (13 \textit {\_a} +90\right )^{{5}/{6}} \sqrt {11 \textit {\_a} -90}}{\left (143 \textit {\_a} +180\right )^{{4}/{3}} \left (20449 \textit {\_a}^{3}-1190700 \textit {\_a} -1458000\right )}d \textit {\_a} +\frac {c_1}{1458000}\right ) x}{\left (\frac {a}{x^{{3}/{4}} \left (y+a \,x^{{1}/{4}}\right )}\right )^{{4}/{3}} x} = 0 \]

✗ Mathematica

ode=y[x]*D[y[x],x]-1/2*a*(x+1)*x^(-7/4)*y[x]==1/4*a^2*(x-1)*(3*x+5)*x^(-5/2); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Not solved

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(-a**2*(x - 1)*(3*x + 5)/(4*x**(5/2)) - a*(x + 1)*y(x)/(2*x**(7/4)) + y(x)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out