ODE
\[ y'(x)^2=a x^n \] ODE Classification
[_quadrature]
Book solution method
Missing Variables ODE, Dependent variable missing, Solve for \(y'\)
Mathematica ✓
cpu = 0.0298183 (sec), leaf count = 57
\[\left \{\left \{y(x)\to c_1-\frac {2 \sqrt {a} x^{\frac {n}{2}+1}}{n+2}\right \},\left \{y(x)\to \frac {2 \sqrt {a} x^{\frac {n}{2}+1}}{n+2}+c_1\right \}\right \}\]
Maple ✓
cpu = 0.446 (sec), leaf count = 41
\[ \left \{ y \left ( x \right ) =-2\,{\frac {x\sqrt {a{x}^{n}}}{2+n}}+{\it \_C1},y \left ( x \right ) =2\,{\frac {x\sqrt {a{x}^{n}}}{2+n}}+{\it \_C1} \right \} \] Mathematica raw input
DSolve[y'[x]^2 == a*x^n,y[x],x]
Mathematica raw output
{{y[x] -> (-2*Sqrt[a]*x^(1 + n/2))/(2 + n) + C[1]}, {y[x] -> (2*Sqrt[a]*x^(1 + n/2))/(2 + n) + C[1]}}
Maple raw input
dsolve(diff(y(x),x)^2 = a*x^n, y(x),'implicit')
Maple raw output
y(x) = 2*x/(2+n)*(a*x^n)^(1/2)+_C1, y(x) = -2*x/(2+n)*(a*x^n)^(1/2)+_C1