4.12.11 \(a x y(x) y'(x)=x^2+y(x)^2\)

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

[[_homogeneous, `class A`], _rational, _Bernoulli]

Book solution method
Homogeneous equation

Mathematica
cpu = 0.0176333 (sec), leaf count = 68

\[\left \{\left \{y(x)\to -\frac {\sqrt {(a-1) c_1 x^{2/a}+x^2}}{\sqrt {a-1}}\right \},\left \{y(x)\to \frac {\sqrt {(a-1) c_1 x^{2/a}+x^2}}{\sqrt {a-1}}\right \}\right \}\]

Maple
cpu = 0.009 (sec), leaf count = 28

\[ \left \{ -{\frac {{x}^{2}}{a-1}}-{x}^{2\,{a}^{-1}}{\it \_C1}+ \left (y \relax (x ) \right ) ^{2}=0 \right \} \] Mathematica raw input

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

Mathematica raw output

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

Maple raw input

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

Maple raw output

-x^2/(a-1)-x^(2/a)*_C1+y(x)^2 = 0