29.9.26 problem 266

Internal problem ID [4866]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 9
Problem number : 266
Date solved : Sunday, March 30, 2025 at 04:07:52 AM
CAS classification : [[_homogeneous, `class G`], _rational, [_Riccati, _special]]

\begin{align*} x^{2} y^{\prime }&=a +b \,x^{2} y^{2} \end{align*}

✓ Maple. Time used: 0.006 (sec). Leaf size: 40

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

\[ y = \frac {-1+\tan \left (\frac {\sqrt {4 a b -1}\, \left (\ln \left (x \right )-c_1 \right )}{2}\right ) \sqrt {4 a b -1}}{2 x b} \]

✓ Mathematica. Time used: 0.193 (sec). Leaf size: 77

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

\begin{align*} y(x)\to \frac {-1+\sqrt {1-4 a b} \left (-1+\frac {2 c_1}{x^{\sqrt {1-4 a b}}+c_1}\right )}{2 b x} \\ y(x)\to \frac {\sqrt {1-4 a b}-1}{2 b x} \\ \end{align*}

✓ Sympy. Time used: 0.221 (sec). Leaf size: 39

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

\[ y{\left (x \right )} = - \frac {- \sqrt {4 a b - 1} \tan {\left (C_{1} + \frac {\sqrt {4 a b - 1} \log {\left (x \right )}}{2} \right )} + 1}{2 b x} \]