ODE
\[ y'(x)^3=a x^n \] ODE Classification
[_quadrature]
Book solution method
Missing Variables ODE, Dependent variable missing, Solve for \(y'\)
Mathematica ✓
cpu = 0.170642 (sec), leaf count = 95
\[\left \{\left \{y(x)\to \frac {3 \sqrt [3]{a} x^{\frac {n}{3}+1}}{n+3}+c_1\right \},\left \{y(x)\to -\frac {3 \sqrt [3]{-1} \sqrt [3]{a} x^{\frac {n}{3}+1}}{n+3}+c_1\right \},\left \{y(x)\to \frac {3 (-1)^{2/3} \sqrt [3]{a} x^{\frac {n}{3}+1}}{n+3}+c_1\right \}\right \}\]
Maple ✓
cpu = 0.28 (sec), leaf count = 77
\[\left [y \left (x \right ) = \frac {3 x \left (a \,x^{n}\right )^{\frac {1}{3}}}{n +3}+\textit {\_C1}, y \left (x \right ) = \frac {3 x \left (-1+i \sqrt {3}\right ) \left (a \,x^{n}\right )^{\frac {1}{3}}}{2 \left (n +3\right )}+\textit {\_C1}, y \left (x \right ) = -\frac {3 x \left (1+i \sqrt {3}\right ) \left (a \,x^{n}\right )^{\frac {1}{3}}}{2 \left (n +3\right )}+\textit {\_C1}\right ]\] 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))
Maple raw output
[y(x) = 3*x/(n+3)*(a*x^n)^(1/3)+_C1, y(x) = 3/2*x/(n+3)*(-1+I*3^(1/2))*(a*x^n)^(1/3)+_C1, y(x) = -3/2*x/(n+3)*(1+I*3^(1/2))*(a*x^n)^(1/3)+_C1]