60.1.102 problem 104

Internal problem ID [10116]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 104
Date solved : Sunday, March 30, 2025 at 03:16:46 PM
CAS classification : [_rational, [_1st_order, `_with_symmetry_[F(x),G(x)]`], _Riccati]

\begin{align*} x y^{\prime }+a x y^{2}+2 y+b x&=0 \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 80

ode:=x*diff(y(x),x)+a*x*y(x)^2+2*y(x)+b*x = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 2.934 (sec). Leaf size: 43

ode=x*D[y[x],x] + a*x*y[x]^2 + 2*y[x] + b*x==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to -\frac {1}{a x}-\sqrt {\frac {b}{a}} \tan \left (a x \sqrt {\frac {b}{a}}-c_1\right ) \]

✗ Sympy

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

NonlinearError : nonlinear term: sqrt(-a*b)