4.21.22 $y^{\prime }(x{)}^3=a{x}^n$

ODE
$$y^{\prime }(x{)}^3=a{x}^n$$ ODE Classification

[_quadrature]

Book solution method
Missing Variables ODE, Dependent variable missing, Solve for $y^{\prime }$

Mathematica ✓
cpu = 0.0137978 (sec), leaf count = 95

$$\{\{y(x)\to \frac{3\sqrt[3]{a}{x}^{\frac{n}{3}+1}}{n+3}+c_1\},\{y(x)\to c_1-\frac{3\sqrt[3]{-1}\sqrt[3]{a}{x}^{\frac{n}{3}+1}}{n+3}\},\{y(x)\to \frac{3(-1{)}^{2/3}\sqrt[3]{a}{x}^{\frac{n}{3}+1}}{n+3}+c_1\}\}$$

Maple ✓
cpu = 0.104 (sec), leaf count = 81

$$\{y\left(x\right)=3\frac{x\sqrt[3]{a{x}^n}}{n+3}+\mathit{\_}\mathit{C}\mathit{1},y\left(x\right)=3\frac{x(i\sqrt{3}-1)\sqrt[3]{a{x}^n}}{2n+6}+\mathit{\_}\mathit{C}\mathit{1},y\left(x\right)=-3\frac{x(i\sqrt{3}+1)\sqrt[3]{a{x}^n}}{2n+6}+\mathit{\_}\mathit{C}\mathit{1}\}$$ 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