4.2.19 \(y'(x)=x^n \left (a+b y(x)^2\right )\)

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

[_separable]

Book solution method
Separable ODE, Neither variable missing

Mathematica
cpu = 0.0569839 (sec), leaf count = 42

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

Maple
cpu = 0.007 (sec), leaf count = 33

\[ \left \{ {\frac {{x}^{n+1}}{n+1}}-{1\arctan \left ({by \relax (x ) {\frac {1}{\sqrt {ab}}}} \right ) {\frac {1}{\sqrt {ab}}}}+{\it \_C1}=0 \right \} \] Mathematica raw input

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

Mathematica raw output

{{y[x] -> (Sqrt[a]*Tan[Sqrt[a]*Sqrt[b]*(x^(1 + n)/(1 + n) + C[1])])/Sqrt[b]}}

Maple raw input

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

Maple raw output

x^(n+1)/(n+1)-1/(a*b)^(1/2)*arctan(y(x)*b/(a*b)^(1/2))+_C1 = 0