##### 4.16.6 $$x^2+y'(x)^2=4 y(x)$$

ODE
$x^2+y'(x)^2=4 y(x)$ ODE Classiﬁcation

[[_homogeneous, class G]]

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

Mathematica
cpu = 2.09524 (sec), leaf count = 242

$\left \{\text {Solve}\left [\frac {x \left (-\sqrt {4 y(x)-x^2}\right )+\left (x^2-2 y(x)\right ) \log \left (x^2-2 y(x)\right )+2 y(x)}{2 \left (x^2-2 y(x)\right )}+\frac {\sqrt {4-\frac {x^2}{y(x)}} \sqrt {y(x)} \tanh ^{-1}\left (\frac {x}{\sqrt {4-\frac {x^2}{y(x)}} \sqrt {y(x)}}\right )}{\sqrt {4 y(x)-x^2}}=c_1,y(x)\right ],\text {Solve}\left [\frac {x \sqrt {4 y(x)-x^2}+\left (x^2-2 y(x)\right ) \log \left (x^2-2 y(x)\right )+2 y(x)}{2 \left (x^2-2 y(x)\right )}=\frac {\sqrt {4-\frac {x^2}{y(x)}} \sqrt {y(x)} \tanh ^{-1}\left (\frac {x}{\sqrt {4-\frac {x^2}{y(x)}} \sqrt {y(x)}}\right )}{\sqrt {4 y(x)-x^2}}+c_1,y(x)\right ]\right \}$

Maple
cpu = 1.08 (sec), leaf count = 141

$\left [y \left (x \right ) = \frac {x^{2}}{2}+\frac {{\mathrm e}^{2 \LambertW \left (\frac {x \sqrt {2}\, {\mathrm e}^{-\frac {\textit {\_C1}}{2}}}{2}\right )+\ln \left (2\right )+\textit {\_C1}}}{4}+\frac {{\mathrm e}^{\LambertW \left (\frac {x \sqrt {2}\, {\mathrm e}^{-\frac {\textit {\_C1}}{2}}}{2}\right )+\frac {\ln \left (2\right )}{2}+\frac {\textit {\_C1}}{2}} x}{2}, y \left (x \right ) = \frac {x^{2} \left (2 \LambertW \left (-\frac {\sqrt {2}\, x \textit {\_C1}}{2}\right )^{2}+2 \LambertW \left (-\frac {\sqrt {2}\, x \textit {\_C1}}{2}\right )+1\right )}{4 \LambertW \left (-\frac {\sqrt {2}\, x \textit {\_C1}}{2}\right )^{2}}, y \left (x \right ) = \frac {x^{2} \left (2 \LambertW \left (\frac {\sqrt {2}\, x \textit {\_C1}}{2}\right )^{2}+2 \LambertW \left (\frac {\sqrt {2}\, x \textit {\_C1}}{2}\right )+1\right )}{4 \LambertW \left (\frac {\sqrt {2}\, x \textit {\_C1}}{2}\right )^{2}}\right ]$ Mathematica raw input

DSolve[x^2 + y'[x]^2 == 4*y[x],y[x],x]

Mathematica raw output

{Solve[(ArcTanh[x/(Sqrt[4 - x^2/y[x]]*Sqrt[y[x]])]*Sqrt[4 - x^2/y[x]]*Sqrt[y[x]]
)/Sqrt[-x^2 + 4*y[x]] + (Log[x^2 - 2*y[x]]*(x^2 - 2*y[x]) + 2*y[x] - x*Sqrt[-x^2
 + 4*y[x]])/(2*(x^2 - 2*y[x])) == C[1], y[x]], Solve[(Log[x^2 - 2*y[x]]*(x^2 - 2
*y[x]) + 2*y[x] + x*Sqrt[-x^2 + 4*y[x]])/(2*(x^2 - 2*y[x])) == C[1] + (ArcTanh[x
/(Sqrt[4 - x^2/y[x]]*Sqrt[y[x]])]*Sqrt[4 - x^2/y[x]]*Sqrt[y[x]])/Sqrt[-x^2 + 4*y
[x]], y[x]]}

Maple raw input

dsolve(diff(y(x),x)^2+x^2 = 4*y(x), y(x))

Maple raw output

[y(x) = 1/2*x^2+1/4*exp(LambertW(1/2*x*2^(1/2)*exp(-1/2*_C1))+1/2*ln(2)+1/2*_C1)
^2+1/2*exp(LambertW(1/2*x*2^(1/2)*exp(-1/2*_C1))+1/2*ln(2)+1/2*_C1)*x, y(x) = 1/
4*x^2*(2*LambertW(-1/2*2^(1/2)*x*_C1)^2+2*LambertW(-1/2*2^(1/2)*x*_C1)+1)/Lamber
tW(-1/2*2^(1/2)*x*_C1)^2, y(x) = 1/4*x^2*(2*LambertW(1/2*2^(1/2)*x*_C1)^2+2*Lamb
ertW(1/2*2^(1/2)*x*_C1)+1)/LambertW(1/2*2^(1/2)*x*_C1)^2]