ODE
\[ (a-x) y'(x)^2-b+y(x) y'(x)=0 \] ODE Classification
[[_1st_order, _with_linear_symmetries], _Clairaut]
Book solution method
Clairaut’s equation and related types, \(f(y-x y', y')=0\)
Mathematica ✓
cpu = 0.584142 (sec), leaf count = 201
\[\left \{\left \{y(x)\to -\frac {\sqrt {b \left (\sinh \left (2 c_1\right )+\cosh \left (2 c_1\right )\right ) \left (\cosh \left (\frac {c_1}{2}\right ) (a+4 b-x-1)-\sinh \left (\frac {c_1}{2}\right ) (a+4 b-x+1)\right ){}^2}+2 a b-8 b^2+2 b \sinh \left (c_1\right )+2 b \cosh \left (c_1\right )-2 b x}{-4 b+\sinh \left (c_1\right )+\cosh \left (c_1\right )}\right \},\left \{y(x)\to -\frac {-\sqrt {b \left (\sinh \left (2 c_1\right )+\cosh \left (2 c_1\right )\right ) \left (\cosh \left (\frac {c_1}{2}\right ) (a+4 b-x-1)-\sinh \left (\frac {c_1}{2}\right ) (a+4 b-x+1)\right ){}^2}+2 a b-8 b^2+2 b \sinh \left (c_1\right )+2 b \cosh \left (c_1\right )-2 b x}{-4 b+\sinh \left (c_1\right )+\cosh \left (c_1\right )}\right \}\right \}\]
Maple ✓
cpu = 0.026 (sec), leaf count = 34
\[ \left \{ \left ( y \left ( x \right ) \right ) ^{2}+4\,b \left ( a-x \right ) =0,y \left ( x \right ) ={\frac { \left ( x-a \right ) {{\it \_C1}}^{2}+b}{{\it \_C1}}} \right \} \] Mathematica raw input
DSolve[-b + y[x]*y'[x] + (a - x)*y'[x]^2 == 0,y[x],x]
Mathematica raw output
{{y[x] -> -((2*a*b - 8*b^2 - 2*b*x + 2*b*Cosh[C[1]] + 2*b*Sinh[C[1]] + Sqrt[b*((-1 + a + 4*b - x)*Cosh[C[1]/2] - (1 + a + 4*b - x)*Sinh[C[1]/2])^2*(Cosh[2*C[1]] + Sinh[2*C[1]])])/(-4*b + Cosh[C[1]] + Sinh[C[1]]))}, {y[x] -> -((2*a*b - 8*b^2 - 2*b*x + 2*b*Cosh[C[1]] + 2*b*Sinh[C[1]] - Sqrt[b*((-1 + a + 4*b - x)*Cosh[C[1]/2] - (1 + a + 4*b - x)*Sinh[C[1]/2])^2*(Cosh[2*C[1]] + Sinh[2*C[1]])])/(-4*b + Cosh[C[1]] + Sinh[C[1]]))}}
Maple raw input
dsolve((a-x)*diff(y(x),x)^2+y(x)*diff(y(x),x)-b = 0, y(x),'implicit')
Maple raw output
y(x)^2+4*b*(a-x) = 0, y(x) = ((x-a)*_C1^2+b)/_C1