ODE
\[ -a y(x) y'(x)-a x+y(x) \sqrt {y'(x)^2+1}=0 \] ODE Classification
[[_homogeneous, `class A`], _rational, _dAlembert]
Book solution method
No Missing Variables ODE, Solve for \(y\)
Mathematica ✓
cpu = 0.588497 (sec), leaf count = 126
\[\left \{\left \{y(x)\to -\frac {\sqrt {\left (a^2-1\right )^3 \left (-x^2\right )+2 \left (a^2-1\right ) x e^{\left (a^2-1\right ) c_1}+e^{2 \left (a^2-1\right ) c_1}}}{\sqrt {\left (a^2-1\right )^3}}\right \},\left \{y(x)\to \frac {\sqrt {\left (a^2-1\right )^3 \left (-x^2\right )+2 \left (a^2-1\right ) x e^{\left (a^2-1\right ) c_1}+e^{2 \left (a^2-1\right ) c_1}}}{\sqrt {\left (a^2-1\right )^3}}\right \}\right \}\]
Maple ✓
cpu = 0.392 (sec), leaf count = 223
\[\left [x -{\mathrm e}^{\int _{}^{\frac {-a^{2} x +\sqrt {a^{2} x^{2}+a^{2} y \left (x \right )^{2}-y \left (x \right )^{2}}}{\left (a^{2}-1\right ) y \left (x \right )}}\frac {\left (a \sqrt {\textit {\_a}^{2}+1}-\textit {\_a} \right ) a}{\sqrt {\textit {\_a}^{2}+1}\, \left (-a \textit {\_a} +\sqrt {\textit {\_a}^{2}+1}\right ) \left (-a \,\textit {\_a}^{2}+\sqrt {\textit {\_a}^{2}+1}\, \textit {\_a} -a \right )}d \textit {\_a}} \textit {\_C1} = 0, x -{\mathrm e}^{\int _{}^{-\frac {a^{2} x +\sqrt {a^{2} x^{2}+a^{2} y \left (x \right )^{2}-y \left (x \right )^{2}}}{\left (a^{2}-1\right ) y \left (x \right )}}\frac {\left (a \sqrt {\textit {\_a}^{2}+1}-\textit {\_a} \right ) a}{\sqrt {\textit {\_a}^{2}+1}\, \left (-a \textit {\_a} +\sqrt {\textit {\_a}^{2}+1}\right ) \left (-a \,\textit {\_a}^{2}+\sqrt {\textit {\_a}^{2}+1}\, \textit {\_a} -a \right )}d \textit {\_a}} \textit {\_C1} = 0\right ]\] Mathematica raw input
DSolve[-(a*x) - a*y[x]*y'[x] + y[x]*Sqrt[1 + y'[x]^2] == 0,y[x],x]
Mathematica raw output
{{y[x] -> -(Sqrt[E^(2*(-1 + a^2)*C[1]) + 2*(-1 + a^2)*E^((-1 + a^2)*C[1])*x - (-1 + a^2)^3*x^2]/Sqrt[(-1 + a^2)^3])}, {y[x] -> Sqrt[E^(2*(-1 + a^2)*C[1]) + 2*(-1 + a^2)*E^((-1 + a^2)*C[1])*x - (-1 + a^2)^3*x^2]/Sqrt[(-1 + a^2)^3]}}
Maple raw input
dsolve(y(x)*(1+diff(y(x),x)^2)^(1/2)-a*y(x)*diff(y(x),x)-a*x = 0, y(x))
Maple raw output
[x-exp(Intat((a*(_a^2+1)^(1/2)-_a)*a/(_a^2+1)^(1/2)/(-a*_a+(_a^2+1)^(1/2))/(-a*_a^2+(_a^2+1)^(1/2)*_a-a),_a = (-a^2*x+(a^2*x^2+a^2*y(x)^2-y(x)^2)^(1/2))/(a^2-1)/y(x)))*_C1 = 0, x-exp(Intat((a*(_a^2+1)^(1/2)-_a)*a/(_a^2+1)^(1/2)/(-a*_a+(_a^2+1)^(1/2))/(-a*_a^2+(_a^2+1)^(1/2)*_a-a),_a = -(a^2*x+(a^2*x^2+a^2*y(x)^2-y(x)^2)^(1/2))/(a^2-1)/y(x)))*_C1 = 0]