2.24.16 Problem 25

Internal problem ID [13580]
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 : 25
Date solved : Friday, December 19, 2025 at 07:24:33 AM
CAS classification : [_rational, [_Abel, `2nd type`, `class B`]]

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

Unknown ode type.

2.24.16.1 ✗ Maple

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

\[ \text {No solution found} \]

Maple step by step

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y \left (x \right ) \left (\frac {d}{d x}y \left (x \right )\right )+\frac {a \left (5 x +1\right ) y \left (x \right )}{2 \sqrt {x}}=a^{2} \left (-x^{2}+1\right ) \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=\frac {-\frac {a \left (5 x +1\right ) y \left (x \right )}{2 \sqrt {x}}+a^{2} \left (-x^{2}+1\right )}{y \left (x \right )} \end {array} \]

2.24.16.2 ✗ Mathematica

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

Not solved

2.24.16.3 ✗ Sympy

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

Timed Out