4.11.15 \(x y(x) y'(x)=a+b y(x)^2\)

ODE
\[ x y(x) y'(x)=a+b y(x)^2 \] ODE Classification

[_separable]

Book solution method
Separable ODE, Neither variable missing

Mathematica
cpu = 0.0156619 (sec), leaf count = 58

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

Maple
cpu = 0.008 (sec), leaf count = 21

\[ \left \{ {\frac {a}{b}}-{x}^{2\,b}{\it \_C1}+ \left (y \relax (x ) \right ) ^{2}=0 \right \} \] Mathematica raw input

DSolve[x*y[x]*y'[x] == a + b*y[x]^2,y[x],x]

Mathematica raw output

{{y[x] -> -(Sqrt[-a + E^(2*b*(C[1] + Log[x]))]/Sqrt[b])}, {y[x] -> Sqrt[-a + E^(
2*b*(C[1] + Log[x]))]/Sqrt[b]}}

Maple raw input

dsolve(x*y(x)*diff(y(x),x) = a+b*y(x)^2, y(x),'implicit')

Maple raw output

a/b-x^(2*b)*_C1+y(x)^2 = 0