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 = 0.306254 (sec), leaf count = 148
\[\left \{\left \{y(x)\to -\frac {\sqrt {e^{-2 c_1} \left (\left (a^2-1\right )^3 \left (-e^{2 c_1}\right ) x^2+2 \left (a^2-1\right ) x e^{\left (a^2+1\right ) c_1}+e^{2 a^2 c_1}\right )}}{\sqrt {\left (a^2-1\right )^3}}\right \},\left \{y(x)\to \frac {\sqrt {e^{-2 c_1} \left (\left (a^2-1\right )^3 \left (-e^{2 c_1}\right ) x^2+2 \left (a^2-1\right ) x e^{\left (a^2+1\right ) c_1}+e^{2 a^2 c_1}\right )}}{\sqrt {\left (a^2-1\right )^3}}\right \}\right \}\]
Maple ✓
cpu = 0.262 (sec), leaf count = 129
\[ \left \{ [x \left ( {\it \_T} \right ) ={{\rm e}^{\int \!{a \left ( -{{\it \_T}{\frac {1}{\sqrt {{{\it \_T}}^{2}+1}}}}+a \right ) \left ( a{\it \_T}-\sqrt {{{\it \_T}}^{2}+1} \right ) ^{-1} \left ( a{{\it \_T}}^{2}-{\it \_T}\,\sqrt {{{\it \_T}}^{2}+1}+a \right ) ^{-1}}\,{\rm d}{\it \_T}}}{\it \_C1},y \left ( {\it \_T} \right ) ={{\it \_C1}\,a{{\rm e}^{\int \!{a \left ( -{{\it \_T}{\frac {1}{\sqrt {{{\it \_T}}^{2}+1}}}}+a \right ) \left ( a{\it \_T}-\sqrt {{{\it \_T}}^{2}+1} \right ) ^{-1} \left ( a{{\it \_T}}^{2}-{\it \_T}\,\sqrt {{{\it \_T}}^{2}+1}+a \right ) ^{-1}}\,{\rm d}{\it \_T}}} \left ( -a{\it \_T}+\sqrt {{{\it \_T}}^{2}+1} \right ) ^{-1}}] \right \} \] 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*a^2*C[1]) + 2*(-1 + a^2)*E^((1 + a^2)*C[1])*x - (-1 + a^2)^3*E^(2*C[1])*x^2)/E^(2*C[1])]/Sqrt[(-1 + a^2)^3])}, {y[x] -> Sqrt[(E^(2*a^2*C[1]) + 2*(-1 + a^2)*E^((1 + a^2)*C[1])*x - (-1 + a^2)^3*E^(2*C[1])*x^2)/E^(2*C[1])]/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),'implicit')
Maple raw output
[x(_T) = exp(Int(a*(-1/(_T^2+1)^(1/2)*_T+a)/(a*_T-(_T^2+1)^(1/2))/(a*_T^2-_T*(_T^2+1)^(1/2)+a),_T))*_C1, y(_T) = a*exp(Int(a*(-1/(_T^2+1)^(1/2)*_T+a)/(a*_T-(_T^2+1)^(1/2))/(a*_T^2-_T*(_T^2+1)^(1/2)+a),_T))*_C1/(-a*_T+(_T^2+1)^(1/2))]