ODE
\[ b x \left (a^2-x^2\right ) y'(x)^2+\left (a^2-x^2\right ) y'(x)^3-b x-y'(x)=0 \] ODE Classification
[_quadrature]
Book solution method
Missing Variables ODE, Dependent variable missing, Solve for \(y'\)
Mathematica ✓
cpu = 0.160767 (sec), leaf count = 64
\[\left \{\left \{y(x)\to -\frac {b x^2}{2}+c_1\right \},\left \{y(x)\to -\tan ^{-1}\left (\frac {x}{\sqrt {a^2-x^2}}\right )+c_1\right \},\left \{y(x)\to \tan ^{-1}\left (\frac {x}{\sqrt {a^2-x^2}}\right )+c_1\right \}\right \}\]
Maple ✓
cpu = 0.102 (sec), leaf count = 52
\[\left [y \left (x \right ) = -\frac {b \,x^{2}}{2}+\textit {\_C1}, y \left (x \right ) = \arctan \left (\frac {x}{\sqrt {a^{2}-x^{2}}}\right )+\textit {\_C1}, y \left (x \right ) = -\arctan \left (\frac {x}{\sqrt {a^{2}-x^{2}}}\right )+\textit {\_C1}\right ]\] Mathematica raw input
DSolve[-(b*x) - y'[x] + b*x*(a^2 - x^2)*y'[x]^2 + (a^2 - x^2)*y'[x]^3 == 0,y[x],x]
Mathematica raw output
{{y[x] -> -1/2*(b*x^2) + C[1]}, {y[x] -> -ArcTan[x/Sqrt[a^2 - x^2]] + C[1]}, {y[x] -> ArcTan[x/Sqrt[a^2 - x^2]] + C[1]}}
Maple raw input
dsolve((a^2-x^2)*diff(y(x),x)^3+b*x*(a^2-x^2)*diff(y(x),x)^2-diff(y(x),x)-b*x = 0, y(x))
Maple raw output
[y(x) = -1/2*b*x^2+_C1, y(x) = arctan(1/(a^2-x^2)^(1/2)*x)+_C1, y(x) = -arctan(1/(a^2-x^2)^(1/2)*x)+_C1]