29.12.20 problem 339

Internal problem ID [4939]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 12
Problem number : 339
Date solved : Sunday, March 30, 2025 at 04:17:12 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: 25.363 (sec). Leaf size: 91

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 \frac {\sqrt {A} \tan \left (\sqrt {A} \sqrt {B} \left (\frac {\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b}}+c_1\right )\right )}{\sqrt {B}} \\ y(x)\to -\frac {i \sqrt {A}}{\sqrt {B}} \\ y(x)\to \frac {i \sqrt {A}}{\sqrt {B}} \\ \end{align*}

✓ Sympy. Time used: 4.987 (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)
 

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