4.16.17 \(y'(x)^2=a^2 y(x)^n\)

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

[_quadrature]

Book solution method
Missing Variables ODE, Independent variable missing, Solve for \(y'\)

Mathematica
cpu = 0.0470141 (sec), leaf count = 68

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

Maple
cpu = 0.035 (sec), leaf count = 51

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

DSolve[y'[x]^2 == a^2*y[x]^n,y[x],x]

Mathematica raw output

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

Maple raw input

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

Maple raw output

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