23.1.356 problem 342 (a)

Internal problem ID [4963]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 1. THE DIFFERENTIAL EQUATION IS OF FIRST ORDER AND OF FIRST DEGREE, page 223
Problem number : 342 (a)
Date solved : Tuesday, September 30, 2025 at 09:06:10 AM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 42

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

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

✓ Mathematica. Time used: 0.254 (sec). Leaf size: 81

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

\begin{align*} y(x)&\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{B K[1]^2+A}dK[1]\&\right ]\left [\int _1^x\frac {1}{b K[2]^2+a}dK[2]+c_1\right ]\\ y(x)&\to -\frac {i \sqrt {A}}{\sqrt {B}}\\ y(x)&\to \frac {i \sqrt {A}}{\sqrt {B}} \end{align*}

✓ Sympy. Time used: 2.999 (sec). Leaf size: 112

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

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