74.6.5 problem 8 (eq 68)

Internal problem ID [19855]
Book : Elementary Differential Equations. By Thornton C. Fry. D Van Nostrand. NY. First Edition (1929)
Section : Chapter IV. Methods of solution: First order equations. section 33. Problems at page 91
Problem number : 8 (eq 68)
Date solved : Thursday, October 02, 2025 at 04:51:16 PM
CAS classification : [_separable]

\begin{align*} y^{\prime }&=x \left (a y^{2}+b \right ) \end{align*}

✓ Maple. Time used: 0.005 (sec). Leaf size: 28

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

\[ y = \frac {\tan \left (\frac {\sqrt {b a}\, \left (x^{2}+2 c_1 \right )}{2}\right ) \sqrt {b a}}{a} \]

✓ Mathematica. Time used: 6.795 (sec). Leaf size: 75

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

\begin{align*} y(x)&\to \frac {\sqrt {b} \tan \left (\frac {1}{2} \sqrt {a} \sqrt {b} \left (x^2+2 c_1\right )\right )}{\sqrt {a}}\\ y(x)&\to -\frac {i \sqrt {b}}{\sqrt {a}}\\ y(x)&\to \frac {i \sqrt {b}}{\sqrt {a}} \end{align*}

✓ Sympy. Time used: 0.919 (sec). Leaf size: 63

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

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