4.16.13 \(y'(x)^2=a+b y(x)^2\)

ODE
\[ y'(x)^2=a+b y(x)^2 \] ODE Classification

[_quadrature]

Book solution method
Missing Variables ODE, Independent variable missing, Solve for \(y'\)

Mathematica
cpu = 0.0973104 (sec), leaf count = 93

\[\left \{\left \{y(x)\to \frac {e^{-\sqrt {b} \left (c_1+x\right )} \left (e^{2 \sqrt {b} \left (c_1+x\right )}-a b\right )}{2 b}\right \},\left \{y(x)\to \frac {e^{-\sqrt {b} \left (c_1+x\right )} \left (e^{2 \sqrt {b} c_1}-a b e^{2 \sqrt {b} x}\right )}{2 b}\right \}\right \}\]

Maple
cpu = 0.036 (sec), leaf count = 72

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

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

Mathematica raw output

{{y[x] -> (-(a*b) + E^(2*Sqrt[b]*(x + C[1])))/(2*b*E^(Sqrt[b]*(x + C[1])))}, {y[
x] -> (-(a*b*E^(2*Sqrt[b]*x)) + E^(2*Sqrt[b]*C[1]))/(2*b*E^(Sqrt[b]*(x + C[1])))
}}

Maple raw input

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

Maple raw output

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