60.1.104 problem 104

Internal problem ID [10118]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 104
Date solved : Monday, January 27, 2025 at 06:28:10 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*}

✓ Solution by Maple

Time used: 0.001 (sec). Leaf size: 80

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

\[ y = \frac {-2 a b c_{1} x -i {\mathrm e}^{-2 i \sqrt {a}\, \sqrt {b}\, x} \sqrt {a}\, \sqrt {b}\, x -2 i c_{1} \sqrt {a}\, \sqrt {b}-{\mathrm e}^{-2 i \sqrt {a}\, \sqrt {b}\, x}}{x a \left (2 i c_{1} \sqrt {a}\, \sqrt {b}+{\mathrm e}^{-2 i \sqrt {a}\, \sqrt {b}\, x}\right )} \]

✓ Solution by Mathematica

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

DSolve[x*D[y[x],x] + a*x*y[x]^2 + 2*y[x] + b*x==0,y[x],x,IncludeSingularSolutions -> True]
 

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