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 = 0.0914421 (sec), leaf count = 110

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

Maple
cpu = 0.052 (sec), leaf count = 136

\[ \left \{ x-{\frac {1}{a}\sqrt {-{a}^{2} \left (y \relax (x ) \right ) ^{2}+ay \relax (x ) }}-{\frac {1}{2}\arctan \left ({1\sqrt {{a}^{2}} \left (y \relax (x ) -{\frac {1}{2\,a}} \right ) {\frac {1}{\sqrt {-{a}^{2} \left (y \relax (x ) \right ) ^{2}+ay \relax (x ) }}}} \right ) {\frac {1}{\sqrt {{a}^{2}}}}}-{\it \_C1}=0,x+{\frac {1}{a}\sqrt {-{a}^{2} \left (y \relax (x ) \right ) ^{2}+ay \relax (x ) }}+{\frac {1}{2}\arctan \left ({1\sqrt {{a}^{2}} \left (y \relax (x ) -{\frac {1}{2\,a}} \right ) {\frac {1}{\sqrt {-{a}^{2} \left (y \relax (x ) \right ) ^{2}+ay \relax (x ) }}}} \right ) {\frac {1}{\sqrt {{a}^{2}}}}}-{\it \_C1}=0,y \relax (x ) =0 \right \} \] 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),'implicit')

Maple raw output

y(x) = 0, x+1/a*(-a^2*y(x)^2+a*y(x))^(1/2)+1/2/(a^2)^(1/2)*arctan((a^2)^(1/2)*(y
(x)-1/2/a)/(-a^2*y(x)^2+a*y(x))^(1/2))-_C1 = 0, x-1/a*(-a^2*y(x)^2+a*y(x))^(1/2)
-1/2/(a^2)^(1/2)*arctan((a^2)^(1/2)*(y(x)-1/2/a)/(-a^2*y(x)^2+a*y(x))^(1/2))-_C1
 = 0