##### 4.20.12 $$(1-a y(x)) y'(x)^2=a y(x)$$

ODE
$(1-a y(x)) y'(x)^2=a y(x)$ ODE Classiﬁcation

[_quadrature]

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

Mathematica
cpu = 0.244822 (sec), leaf count = 110

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

Maple
cpu = 4.331 (sec), leaf count = 815

$\left [y \left (x \right ) = 0, y \left (x \right ) = \frac {\RootOf \left (4 a^{2} \textit {\_C1}^{2}-8 a^{2} \textit {\_C1} x +4 a^{2} x^{2}-4 \sqrt {a^{2}}\, \textit {\_C1} \RootOf \left (4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) \textit {\_C1}^{2} a^{2}-8 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) \textit {\_C1} \,a^{2} x +4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) a^{2} x^{2}-4 \sqrt {a^{2}}\, \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) \textit {\_C1} \textit {\_Z} +4 \sqrt {a^{2}}\, \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) \textit {\_Z} x +\left (\tan ^{2}\left (\textit {\_Z} \right )\right ) \textit {\_Z}^{2}+4 a^{2} \textit {\_C1}^{2}-8 a^{2} \textit {\_C1} x +4 a^{2} x^{2}-4 \sqrt {a^{2}}\, \textit {\_C1} \textit {\_Z} +4 \sqrt {a^{2}}\, x \textit {\_Z} +\textit {\_Z}^{2}-1\right )+4 \sqrt {a^{2}}\, x \RootOf \left (4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) \textit {\_C1}^{2} a^{2}-8 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) \textit {\_C1} \,a^{2} x +4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) a^{2} x^{2}-4 \sqrt {a^{2}}\, \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) \textit {\_C1} \textit {\_Z} +4 \sqrt {a^{2}}\, \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) \textit {\_Z} x +\left (\tan ^{2}\left (\textit {\_Z} \right )\right ) \textit {\_Z}^{2}+4 a^{2} \textit {\_C1}^{2}-8 a^{2} \textit {\_C1} x +4 a^{2} x^{2}-4 \sqrt {a^{2}}\, \textit {\_C1} \textit {\_Z} +4 \sqrt {a^{2}}\, x \textit {\_Z} +\textit {\_Z}^{2}-1\right )+\RootOf \left (4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) \textit {\_C1}^{2} a^{2}-8 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) \textit {\_C1} \,a^{2} x +4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) a^{2} x^{2}-4 \sqrt {a^{2}}\, \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) \textit {\_C1} \textit {\_Z} +4 \sqrt {a^{2}}\, \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) \textit {\_Z} x +\left (\tan ^{2}\left (\textit {\_Z} \right )\right ) \textit {\_Z}^{2}+4 a^{2} \textit {\_C1}^{2}-8 a^{2} \textit {\_C1} x +4 a^{2} x^{2}-4 \sqrt {a^{2}}\, \textit {\_C1} \textit {\_Z} +4 \sqrt {a^{2}}\, x \textit {\_Z} +\textit {\_Z}^{2}-1\right )^{2}+\textit {\_Z}^{2}-2 \textit {\_Z} \right )}{2 a}, y \left (x \right ) = \frac {\RootOf \left (4 a^{2} \textit {\_C1}^{2}-8 a^{2} \textit {\_C1} x +4 a^{2} x^{2}+4 \sqrt {a^{2}}\, \textit {\_C1} \RootOf \left (4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) \textit {\_C1}^{2} a^{2}-8 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) \textit {\_C1} \,a^{2} x +4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) a^{2} x^{2}+4 \sqrt {a^{2}}\, \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) \textit {\_C1} \textit {\_Z} -4 \sqrt {a^{2}}\, \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) \textit {\_Z} x +\left (\tan ^{2}\left (\textit {\_Z} \right )\right ) \textit {\_Z}^{2}+4 a^{2} \textit {\_C1}^{2}-8 a^{2} \textit {\_C1} x +4 a^{2} x^{2}+4 \sqrt {a^{2}}\, \textit {\_C1} \textit {\_Z} -4 \sqrt {a^{2}}\, x \textit {\_Z} +\textit {\_Z}^{2}-1\right )-4 \sqrt {a^{2}}\, x \RootOf \left (4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) \textit {\_C1}^{2} a^{2}-8 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) \textit {\_C1} \,a^{2} x +4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) a^{2} x^{2}+4 \sqrt {a^{2}}\, \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) \textit {\_C1} \textit {\_Z} -4 \sqrt {a^{2}}\, \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) \textit {\_Z} x +\left (\tan ^{2}\left (\textit {\_Z} \right )\right ) \textit {\_Z}^{2}+4 a^{2} \textit {\_C1}^{2}-8 a^{2} \textit {\_C1} x +4 a^{2} x^{2}+4 \sqrt {a^{2}}\, \textit {\_C1} \textit {\_Z} -4 \sqrt {a^{2}}\, x \textit {\_Z} +\textit {\_Z}^{2}-1\right )+\RootOf \left (4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) \textit {\_C1}^{2} a^{2}-8 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) \textit {\_C1} \,a^{2} x +4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) a^{2} x^{2}+4 \sqrt {a^{2}}\, \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) \textit {\_C1} \textit {\_Z} -4 \sqrt {a^{2}}\, \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) \textit {\_Z} x +\left (\tan ^{2}\left (\textit {\_Z} \right )\right ) \textit {\_Z}^{2}+4 a^{2} \textit {\_C1}^{2}-8 a^{2} \textit {\_C1} x +4 a^{2} x^{2}+4 \sqrt {a^{2}}\, \textit {\_C1} \textit {\_Z} -4 \sqrt {a^{2}}\, x \textit {\_Z} +\textit {\_Z}^{2}-1\right )^{2}+\textit {\_Z}^{2}-2 \textit {\_Z} \right )}{2 a}\right ]$ Mathematica raw input

DSolve[(1 - a*y[x])*y'[x]^2 == a*y[x],y[x],x]

Mathematica raw output

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

Maple raw input

dsolve((1-a*y(x))*diff(y(x),x)^2 = a*y(x), y(x))

Maple raw output

[y(x) = 0, y(x) = 1/2*RootOf(4*a^2*_C1^2-8*a^2*_C1*x+4*a^2*x^2-4*(a^2)^(1/2)*_C1
*RootOf(4*tan(_Z)^2*_C1^2*a^2-8*tan(_Z)^2*_C1*a^2*x+4*tan(_Z)^2*a^2*x^2-4*(a^2)^
(1/2)*tan(_Z)^2*_C1*_Z+4*(a^2)^(1/2)*tan(_Z)^2*_Z*x+tan(_Z)^2*_Z^2+4*a^2*_C1^2-8
*a^2*_C1*x+4*a^2*x^2-4*(a^2)^(1/2)*_C1*_Z+4*(a^2)^(1/2)*x*_Z+_Z^2-1)+4*(a^2)^(1/
2)*x*RootOf(4*tan(_Z)^2*_C1^2*a^2-8*tan(_Z)^2*_C1*a^2*x+4*tan(_Z)^2*a^2*x^2-4*(a
^2)^(1/2)*tan(_Z)^2*_C1*_Z+4*(a^2)^(1/2)*tan(_Z)^2*_Z*x+tan(_Z)^2*_Z^2+4*a^2*_C1
^2-8*a^2*_C1*x+4*a^2*x^2-4*(a^2)^(1/2)*_C1*_Z+4*(a^2)^(1/2)*x*_Z+_Z^2-1)+RootOf(
4*tan(_Z)^2*_C1^2*a^2-8*tan(_Z)^2*_C1*a^2*x+4*tan(_Z)^2*a^2*x^2-4*(a^2)^(1/2)*ta
n(_Z)^2*_C1*_Z+4*(a^2)^(1/2)*tan(_Z)^2*_Z*x+tan(_Z)^2*_Z^2+4*a^2*_C1^2-8*a^2*_C1
*x+4*a^2*x^2-4*(a^2)^(1/2)*_C1*_Z+4*(a^2)^(1/2)*x*_Z+_Z^2-1)^2+_Z^2-2*_Z)/a, y(x
) = 1/2*RootOf(4*a^2*_C1^2-8*a^2*_C1*x+4*a^2*x^2+4*(a^2)^(1/2)*_C1*RootOf(4*tan(
_Z)^2*_C1^2*a^2-8*tan(_Z)^2*_C1*a^2*x+4*tan(_Z)^2*a^2*x^2+4*(a^2)^(1/2)*tan(_Z)^
2*_C1*_Z-4*(a^2)^(1/2)*tan(_Z)^2*_Z*x+tan(_Z)^2*_Z^2+4*a^2*_C1^2-8*a^2*_C1*x+4*a
^2*x^2+4*(a^2)^(1/2)*_C1*_Z-4*(a^2)^(1/2)*x*_Z+_Z^2-1)-4*(a^2)^(1/2)*x*RootOf(4*
tan(_Z)^2*_C1^2*a^2-8*tan(_Z)^2*_C1*a^2*x+4*tan(_Z)^2*a^2*x^2+4*(a^2)^(1/2)*tan(
_Z)^2*_C1*_Z-4*(a^2)^(1/2)*tan(_Z)^2*_Z*x+tan(_Z)^2*_Z^2+4*a^2*_C1^2-8*a^2*_C1*x
+4*a^2*x^2+4*(a^2)^(1/2)*_C1*_Z-4*(a^2)^(1/2)*x*_Z+_Z^2-1)+RootOf(4*tan(_Z)^2*_C
1^2*a^2-8*tan(_Z)^2*_C1*a^2*x+4*tan(_Z)^2*a^2*x^2+4*(a^2)^(1/2)*tan(_Z)^2*_C1*_Z
-4*(a^2)^(1/2)*tan(_Z)^2*_Z*x+tan(_Z)^2*_Z^2+4*a^2*_C1^2-8*a^2*_C1*x+4*a^2*x^2+4
*(a^2)^(1/2)*_C1*_Z-4*(a^2)^(1/2)*x*_Z+_Z^2-1)^2+_Z^2-2*_Z)/a]