4.23.36 $-ay(x)y^{\prime }(x)-ax+y(x)\sqrt{y^{\prime }(x{)}^2+1}=0$

ODE
$$-ay(x)y^{\prime }(x)-ax+y(x)\sqrt{y^{\prime }(x{)}^2+1}=0$$ 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

$$\{\{y(x)\to -\frac{\sqrt{e^{-2c_1}({(a^2-1)}^3(-e^{2c_1})x^2+2(a^2-1)x{e}^{(a^2+1)c_1}+e^{2a^2{c}_1})}}{\sqrt{{(a^2-1)}^3}}\},\{y(x)\to \frac{\sqrt{e^{-2c_1}({(a^2-1)}^3(-e^{2c_1})x^2+2(a^2-1)x{e}^{(a^2+1)c_1}+e^{2a^2{c}_1})}}{\sqrt{{(a^2-1)}^3}}\}\}$$

Maple ✓
cpu = 0.262 (sec), leaf count = 129

$$\{[x\left(\mathit{\_}\mathit{T}\right)={\mathrm{e}}^{\int a(-\mathit{\_}\mathit{T}\frac{1}{\sqrt{{\mathit{\_}\mathit{T}}^2+1}}+a){(a\mathit{\_}\mathit{T}-\sqrt{{\mathit{\_}\mathit{T}}^2+1})}^{-1}{(a{\mathit{\_}\mathit{T}}^2-\mathit{\_}\mathit{T}\sqrt{{\mathit{\_}\mathit{T}}^2+1}+a)}^{-1}\mathrm{d}\mathit{\_}\mathit{T}}\mathit{\_}\mathit{C}\mathit{1},y\left(\mathit{\_}\mathit{T}\right)=\mathit{\_}\mathit{C}\mathit{1}a{\mathrm{e}}^{\int a(-\mathit{\_}\mathit{T}\frac{1}{\sqrt{{\mathit{\_}\mathit{T}}^2+1}}+a){(a\mathit{\_}\mathit{T}-\sqrt{{\mathit{\_}\mathit{T}}^2+1})}^{-1}{(a{\mathit{\_}\mathit{T}}^2-\mathit{\_}\mathit{T}\sqrt{{\mathit{\_}\mathit{T}}^2+1}+a)}^{-1}\mathrm{d}\mathit{\_}\mathit{T}}{(-a\mathit{\_}\mathit{T}+\sqrt{{\mathit{\_}\mathit{T}}^2+1})}^{-1}]\}$$ 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))]