##### 4.19.45 $$y(x) y'(x)^2=a$$

ODE
$y(x) y'(x)^2=a$ ODE Classiﬁcation

[_quadrature]

Book solution method
Missing Variables ODE, Independent variable missing, Solve for $$y'$$

Mathematica
cpu = 0.188213 (sec), leaf count = 54

$\left \{\left \{y(x)\to \left (\frac {3}{2}\right )^{2/3} \left (-\sqrt {a} x+c_1\right ){}^{2/3}\right \},\left \{y(x)\to \left (\frac {3}{2}\right )^{2/3} \left (\sqrt {a} x+c_1\right ){}^{2/3}\right \}\right \}$

Maple
cpu = 0.434 (sec), leaf count = 239

$\left [y \left (x \right ) = \frac {\left (-12 \textit {\_C1} \,a^{2}+12 a^{2} x \right )^{\frac {2}{3}}}{4 a}, y \left (x \right ) = \frac {\left (-\frac {\left (-12 \textit {\_C1} \,a^{2}+12 a^{2} x \right )^{\frac {1}{3}}}{4}-\frac {i \sqrt {3}\, \left (-12 \textit {\_C1} \,a^{2}+12 a^{2} x \right )^{\frac {1}{3}}}{4}\right )^{2}}{a}, y \left (x \right ) = \frac {\left (-\frac {\left (-12 \textit {\_C1} \,a^{2}+12 a^{2} x \right )^{\frac {1}{3}}}{4}+\frac {i \sqrt {3}\, \left (-12 \textit {\_C1} \,a^{2}+12 a^{2} x \right )^{\frac {1}{3}}}{4}\right )^{2}}{a}, y \left (x \right ) = \frac {\left (12 \textit {\_C1} \,a^{2}-12 a^{2} x \right )^{\frac {2}{3}}}{4 a}, y \left (x \right ) = \frac {\left (-\frac {\left (12 \textit {\_C1} \,a^{2}-12 a^{2} x \right )^{\frac {1}{3}}}{4}-\frac {i \sqrt {3}\, \left (12 \textit {\_C1} \,a^{2}-12 a^{2} x \right )^{\frac {1}{3}}}{4}\right )^{2}}{a}, y \left (x \right ) = \frac {\left (-\frac {\left (12 \textit {\_C1} \,a^{2}-12 a^{2} x \right )^{\frac {1}{3}}}{4}+\frac {i \sqrt {3}\, \left (12 \textit {\_C1} \,a^{2}-12 a^{2} x \right )^{\frac {1}{3}}}{4}\right )^{2}}{a}\right ]$ Mathematica raw input

DSolve[y[x]*y'[x]^2 == a,y[x],x]

Mathematica raw output

{{y[x] -> (3/2)^(2/3)*(-(Sqrt[a]*x) + C[1])^(2/3)}, {y[x] -> (3/2)^(2/3)*(Sqrt[a
]*x + C[1])^(2/3)}}

Maple raw input

dsolve(y(x)*diff(y(x),x)^2 = a, y(x))

Maple raw output

[y(x) = 1/4*(-12*_C1*a^2+12*a^2*x)^(2/3)/a, y(x) = (-1/4*(-12*_C1*a^2+12*a^2*x)^
(1/3)-1/4*I*3^(1/2)*(-12*_C1*a^2+12*a^2*x)^(1/3))^2/a, y(x) = (-1/4*(-12*_C1*a^2
+12*a^2*x)^(1/3)+1/4*I*3^(1/2)*(-12*_C1*a^2+12*a^2*x)^(1/3))^2/a, y(x) = 1/4*(12
*_C1*a^2-12*a^2*x)^(2/3)/a, y(x) = (-1/4*(12*_C1*a^2-12*a^2*x)^(1/3)-1/4*I*3^(1/
2)*(12*_C1*a^2-12*a^2*x)^(1/3))^2/a, y(x) = (-1/4*(12*_C1*a^2-12*a^2*x)^(1/3)+1/
4*I*3^(1/2)*(12*_C1*a^2-12*a^2*x)^(1/3))^2/a]