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 = 0.103926 (sec), leaf count = 101

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

Maple
cpu = 0.045 (sec), leaf count = 87

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

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

Mathematica raw output

{{y[x] -> InverseFunction[a*ArcTan[Sqrt[#1]/Sqrt[a - #1]] - Sqrt[a - #1]*Sqrt[#1
] & ][-x + C[1]]}, {y[x] -> InverseFunction[a*ArcTan[Sqrt[#1]/Sqrt[a - #1]] - Sq
rt[a - #1]*Sqrt[#1] & ][x + C[1]]}}

Maple raw input

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

Maple raw output

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