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 = 0.673638 (sec), leaf count = 158

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

Maple
cpu = 0.137 (sec), leaf count = 212

\[ \left \{ -{x \left (\sqrt {4\,y \relax (x ) -{x}^{2}}+x \right ) ^{-1}}-\ln \left (\sqrt {4\,y \relax (x ) -{x}^{2}}+x \right ) +{x \left (x-\sqrt {4\,y \relax (x ) -{x}^{2}} \right ) ^{-1}}+\ln \left (-x+\sqrt {4\,y \relax (x ) -{x}^{2}} \right ) -{\frac {{x}^{2}}{{x}^{2}-2\,y \relax (x ) }}-\ln \left (-{x}^{2}+2\,y \relax (x ) \right ) -{\it \_C1}=0,-{x \left (\sqrt {4\,y \relax (x ) -{x}^{2}}+x \right ) ^{-1}}-\ln \left (\sqrt {4\,y \relax (x ) -{x}^{2}}+x \right ) +{x \left (x-\sqrt {4\,y \relax (x ) -{x}^{2}} \right ) ^{-1}}+\ln \left (-x+\sqrt {4\,y \relax (x ) -{x}^{2}} \right ) +{\frac {{x}^{2}}{{x}^{2}-2\,y \relax (x ) }}+\ln \left (-{x}^{2}+2\,y \relax (x ) \right ) -{\it \_C1}=0 \right \} \] Mathematica raw input

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

Mathematica raw output

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

Maple raw input

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

Maple raw output

-x/((4*y(x)-x^2)^(1/2)+x)-ln((4*y(x)-x^2)^(1/2)+x)+x/(x-(4*y(x)-x^2)^(1/2))+ln(-
x+(4*y(x)-x^2)^(1/2))+x^2/(x^2-2*y(x))+ln(-x^2+2*y(x))-_C1 = 0, -x/((4*y(x)-x^2)
^(1/2)+x)-ln((4*y(x)-x^2)^(1/2)+x)+x/(x-(4*y(x)-x^2)^(1/2))+ln(-x+(4*y(x)-x^2)^(
1/2))-x^2/(x^2-2*y(x))-ln(-x^2+2*y(x))-_C1 = 0