##### 4.19.46 $$y(x) y'(x)^2=a^2 x$$

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

[[_homogeneous, class A], _dAlembert]

Book solution method
No Missing Variables ODE, Solve for $$y'$$

Mathematica
cpu = 0.195618 (sec), leaf count = 46

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

Maple
cpu = 0.196 (sec), leaf count = 74

$\left [-\frac {\textit {\_C1} x}{y \left (x \right ) \left (\frac {a^{2} \left (\left (x y \left (x \right )\right )^{\frac {3}{2}} a -y \left (x \right )^{3}\right )}{y \left (x \right )^{3}}\right )^{\frac {2}{3}}}+x = 0, -\frac {\textit {\_C1} x}{y \left (x \right ) \left (-\frac {a^{2} \left (\left (x y \left (x \right )\right )^{\frac {3}{2}} a +y \left (x \right )^{3}\right )}{y \left (x \right )^{3}}\right )^{\frac {2}{3}}}+x = 0\right ]$ Mathematica raw input

DSolve[y[x]*y'[x]^2 == a^2*x,y[x],x]

Mathematica raw output

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

Maple raw input

dsolve(y(x)*diff(y(x),x)^2 = a^2*x, y(x))

Maple raw output

[-_C1/y(x)*x/(a^2*((x*y(x))^(3/2)*a-y(x)^3)/y(x)^3)^(2/3)+x = 0, -_C1/y(x)*x/(-a
^2*((x*y(x))^(3/2)*a+y(x)^3)/y(x)^3)^(2/3)+x = 0]