4.20.7 $$y(x) y'(x)^2+y(x)=a$$

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

[_quadrature]

Book solution method
Missing Variables ODE, Independent variable missing, Use new variable

Mathematica
cpu = 0.304924 (sec), leaf count = 137

$\left \{\left \{y(x)\to \text {InverseFunction}\left [\frac {a^{3/2} \sqrt {1-\frac {\text {\#1}}{a}} \sin ^{-1}\left (\frac {\sqrt {\text {\#1}}}{\sqrt {a}}\right )+\sqrt {\text {\#1}} (\text {\#1}-a)}{\sqrt {a-\text {\#1}}}\& \right ][-x+c_1]\right \},\left \{y(x)\to \text {InverseFunction}\left [\frac {a^{3/2} \sqrt {1-\frac {\text {\#1}}{a}} \sin ^{-1}\left (\frac {\sqrt {\text {\#1}}}{\sqrt {a}}\right )+\sqrt {\text {\#1}} (\text {\#1}-a)}{\sqrt {a-\text {\#1}}}\& \right ][x+c_1]\right \}\right \}$

Maple
cpu = 3.067 (sec), leaf count = 827

$\left [y \left (x \right ) = a, y \left (x \right ) = \frac {\tan \left (\RootOf \left (\left (\tan ^{2}\left (\textit {\_Z} \right )\right ) a^{2} \textit {\_Z}^{2}+4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) \textit {\_C1} a \textit {\_Z} -4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) a x \textit {\_Z} +4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) \textit {\_C1}^{2}-8 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) \textit {\_C1} x +4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) x^{2}+a^{2} \textit {\_Z}^{2}+4 \textit {\_C1} \textit {\_Z} a -4 a x \textit {\_Z} +4 \textit {\_C1}^{2}-8 x \textit {\_C1} -a^{2}+4 x^{2}\right )\right ) \left (-\RootOf \left (\left (\tan ^{2}\left (\textit {\_Z} \right )\right ) a^{2} \textit {\_Z}^{2}+4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) \textit {\_C1} a \textit {\_Z} -4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) a x \textit {\_Z} +4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) \textit {\_C1}^{2}-8 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) \textit {\_C1} x +4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) x^{2}+a^{2} \textit {\_Z}^{2}+4 \textit {\_C1} \textit {\_Z} a -4 a x \textit {\_Z} +4 \textit {\_C1}^{2}-8 x \textit {\_C1} -a^{2}+4 x^{2}\right ) a -2 \textit {\_C1} +2 x \right )}{2}+\frac {a}{2}, y \left (x \right ) = \frac {\tan \left (\RootOf \left (\left (\tan ^{2}\left (\textit {\_Z} \right )\right ) a^{2} \textit {\_Z}^{2}+4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) \textit {\_C1} a \textit {\_Z} -4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) a x \textit {\_Z} +4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) \textit {\_C1}^{2}-8 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) \textit {\_C1} x +4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) x^{2}+a^{2} \textit {\_Z}^{2}+4 \textit {\_C1} \textit {\_Z} a -4 a x \textit {\_Z} +4 \textit {\_C1}^{2}-8 x \textit {\_C1} -a^{2}+4 x^{2}\right )\right ) \left (\RootOf \left (\left (\tan ^{2}\left (\textit {\_Z} \right )\right ) a^{2} \textit {\_Z}^{2}+4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) \textit {\_C1} a \textit {\_Z} -4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) a x \textit {\_Z} +4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) \textit {\_C1}^{2}-8 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) \textit {\_C1} x +4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) x^{2}+a^{2} \textit {\_Z}^{2}+4 \textit {\_C1} \textit {\_Z} a -4 a x \textit {\_Z} +4 \textit {\_C1}^{2}-8 x \textit {\_C1} -a^{2}+4 x^{2}\right ) a +2 \textit {\_C1} -2 x \right )}{2}+\frac {a}{2}, y \left (x \right ) = \frac {\tan \left (\RootOf \left (\left (\tan ^{2}\left (\textit {\_Z} \right )\right ) a^{2} \textit {\_Z}^{2}-4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) \textit {\_C1} a \textit {\_Z} +4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) a x \textit {\_Z} +4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) \textit {\_C1}^{2}-8 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) \textit {\_C1} x +4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) x^{2}+a^{2} \textit {\_Z}^{2}-4 \textit {\_C1} \textit {\_Z} a +4 a x \textit {\_Z} +4 \textit {\_C1}^{2}-8 x \textit {\_C1} -a^{2}+4 x^{2}\right )\right ) \left (-\RootOf \left (\left (\tan ^{2}\left (\textit {\_Z} \right )\right ) a^{2} \textit {\_Z}^{2}-4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) \textit {\_C1} a \textit {\_Z} +4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) a x \textit {\_Z} +4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) \textit {\_C1}^{2}-8 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) \textit {\_C1} x +4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) x^{2}+a^{2} \textit {\_Z}^{2}-4 \textit {\_C1} \textit {\_Z} a +4 a x \textit {\_Z} +4 \textit {\_C1}^{2}-8 x \textit {\_C1} -a^{2}+4 x^{2}\right ) a +2 \textit {\_C1} -2 x \right )}{2}+\frac {a}{2}, y \left (x \right ) = \frac {\tan \left (\RootOf \left (\left (\tan ^{2}\left (\textit {\_Z} \right )\right ) a^{2} \textit {\_Z}^{2}-4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) \textit {\_C1} a \textit {\_Z} +4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) a x \textit {\_Z} +4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) \textit {\_C1}^{2}-8 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) \textit {\_C1} x +4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) x^{2}+a^{2} \textit {\_Z}^{2}-4 \textit {\_C1} \textit {\_Z} a +4 a x \textit {\_Z} +4 \textit {\_C1}^{2}-8 x \textit {\_C1} -a^{2}+4 x^{2}\right )\right ) \left (\RootOf \left (\left (\tan ^{2}\left (\textit {\_Z} \right )\right ) a^{2} \textit {\_Z}^{2}-4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) \textit {\_C1} a \textit {\_Z} +4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) a x \textit {\_Z} +4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) \textit {\_C1}^{2}-8 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) \textit {\_C1} x +4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) x^{2}+a^{2} \textit {\_Z}^{2}-4 \textit {\_C1} \textit {\_Z} a +4 a x \textit {\_Z} +4 \textit {\_C1}^{2}-8 x \textit {\_C1} -a^{2}+4 x^{2}\right ) a -2 \textit {\_C1} +2 x \right )}{2}+\frac {a}{2}\right ]$ Mathematica raw input

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

Mathematica raw output

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

Maple raw input

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

Maple raw output

[y(x) = a, y(x) = 1/2*tan(RootOf(tan(_Z)^2*a^2*_Z^2+4*tan(_Z)^2*_C1*a*_Z-4*tan(_
Z)^2*a*x*_Z+4*tan(_Z)^2*_C1^2-8*tan(_Z)^2*_C1*x+4*tan(_Z)^2*x^2+a^2*_Z^2+4*_C1*_
Z*a-4*a*x*_Z+4*_C1^2-8*x*_C1-a^2+4*x^2))*(-RootOf(tan(_Z)^2*a^2*_Z^2+4*tan(_Z)^2
*_C1*a*_Z-4*tan(_Z)^2*a*x*_Z+4*tan(_Z)^2*_C1^2-8*tan(_Z)^2*_C1*x+4*tan(_Z)^2*x^2
+a^2*_Z^2+4*_C1*_Z*a-4*a*x*_Z+4*_C1^2-8*x*_C1-a^2+4*x^2)*a-2*_C1+2*x)+1/2*a, y(x
) = 1/2*tan(RootOf(tan(_Z)^2*a^2*_Z^2+4*tan(_Z)^2*_C1*a*_Z-4*tan(_Z)^2*a*x*_Z+4*
tan(_Z)^2*_C1^2-8*tan(_Z)^2*_C1*x+4*tan(_Z)^2*x^2+a^2*_Z^2+4*_C1*_Z*a-4*a*x*_Z+4
*_C1^2-8*x*_C1-a^2+4*x^2))*(RootOf(tan(_Z)^2*a^2*_Z^2+4*tan(_Z)^2*_C1*a*_Z-4*tan
(_Z)^2*a*x*_Z+4*tan(_Z)^2*_C1^2-8*tan(_Z)^2*_C1*x+4*tan(_Z)^2*x^2+a^2*_Z^2+4*_C1
*_Z*a-4*a*x*_Z+4*_C1^2-8*x*_C1-a^2+4*x^2)*a+2*_C1-2*x)+1/2*a, y(x) = 1/2*tan(Roo
tOf(tan(_Z)^2*a^2*_Z^2-4*tan(_Z)^2*_C1*a*_Z+4*tan(_Z)^2*a*x*_Z+4*tan(_Z)^2*_C1^2
-8*tan(_Z)^2*_C1*x+4*tan(_Z)^2*x^2+a^2*_Z^2-4*_C1*_Z*a+4*a*x*_Z+4*_C1^2-8*x*_C1-
a^2+4*x^2))*(-RootOf(tan(_Z)^2*a^2*_Z^2-4*tan(_Z)^2*_C1*a*_Z+4*tan(_Z)^2*a*x*_Z+
4*tan(_Z)^2*_C1^2-8*tan(_Z)^2*_C1*x+4*tan(_Z)^2*x^2+a^2*_Z^2-4*_C1*_Z*a+4*a*x*_Z
+4*_C1^2-8*x*_C1-a^2+4*x^2)*a+2*_C1-2*x)+1/2*a, y(x) = 1/2*tan(RootOf(tan(_Z)^2*
a^2*_Z^2-4*tan(_Z)^2*_C1*a*_Z+4*tan(_Z)^2*a*x*_Z+4*tan(_Z)^2*_C1^2-8*tan(_Z)^2*_
C1*x+4*tan(_Z)^2*x^2+a^2*_Z^2-4*_C1*_Z*a+4*a*x*_Z+4*_C1^2-8*x*_C1-a^2+4*x^2))*(R
ootOf(tan(_Z)^2*a^2*_Z^2-4*tan(_Z)^2*_C1*a*_Z+4*tan(_Z)^2*a*x*_Z+4*tan(_Z)^2*_C1
^2-8*tan(_Z)^2*_C1*x+4*tan(_Z)^2*x^2+a^2*_Z^2-4*_C1*_Z*a+4*a*x*_Z+4*_C1^2-8*x*_C
1-a^2+4*x^2)*a-2*_C1+2*x)+1/2*a]