ODE
\[ y'(x)^3=a x^n \] ODE Classification
[_quadrature]
Book solution method
Missing Variables ODE, Dependent variable missing, Solve for \(y'\)
Mathematica ✓
cpu = 0.0137978 (sec), leaf count = 95
\[\left \{\left \{y(x)\to \frac {3 \sqrt [3]{a} x^{\frac {n}{3}+1}}{n+3}+c_1\right \},\left \{y(x)\to c_1-\frac {3 \sqrt [3]{-1} \sqrt [3]{a} x^{\frac {n}{3}+1}}{n+3}\right \},\left \{y(x)\to \frac {3 (-1)^{2/3} \sqrt [3]{a} x^{\frac {n}{3}+1}}{n+3}+c_1\right \}\right \}\]
Maple ✓
cpu = 0.104 (sec), leaf count = 81
\[ \left \{ y \left ( x \right ) =3\,{\frac {x\sqrt [3]{a{x}^{n}}}{n+3}}+{\it \_C1},y \left ( x \right ) =3\,{\frac {x \left ( i\sqrt {3}-1 \right ) \sqrt [3]{a{x}^{n}}}{2\,n+6}}+{\it \_C1},y \left ( x \right ) =-3\,{\frac {x \left ( i\sqrt {3}+1 \right ) \sqrt [3]{a{x}^{n}}}{2\,n+6}}+{\it \_C1} \right \} \] Mathematica raw input
DSolve[y'[x]^3 == a*x^n,y[x],x]
Mathematica raw output
{{y[x] -> (3*a^(1/3)*x^(1 + n/3))/(3 + n) + C[1]}, {y[x] -> (-3*(-1)^(1/3)*a^(1/3)*x^(1 + n/3))/(3 + n) + C[1]}, {y[x] -> (3*(-1)^(2/3)*a^(1/3)*x^(1 + n/3))/(3 + n) + C[1]}}
Maple raw input
dsolve(diff(y(x),x)^3 = a*x^n, y(x),'implicit')
Maple raw output
y(x) = 3*x/(n+3)*(a*x^n)^(1/3)+_C1, y(x) = 3*x*(I*3^(1/2)-1)*(a*x^n)^(1/3)/(2*n+6)+_C1, y(x) = -3*x*(I*3^(1/2)+1)*(a*x^n)^(1/3)/(2*n+6)+_C1