4.23.35 $ax\sqrt{y^{\prime }(x{)}^2+1}+x{y}^{\prime }(x)-y(x)=0$

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

[[_homogeneous, `class A`], _dAlembert]

Book solution method
No Missing Variables ODE, Solve for $y$

Mathematica ✓
cpu = 0.789498 (sec), leaf count = 369

$$\{\text{Solve}[\frac{a(2\log (x-a^{2}x)-\log \left(\frac{(a^2-1)(y(x)+ix(a\sqrt{a^2-\frac{y(x{)}^2}{x^2}-1}+a^2-1))}{a^3(x+iy(x))}\right)+\log \left(\frac{i(a^2-1)(x(a\sqrt{a^2-\frac{y(x{)}^2}{x^2}-1}+a^2-1)+iy(x))}{a^3(x-iy(x))}\right)+\log (\frac{y(x{)}^2}{x^2}+1))-2i{\tan }^{-1}\left(\frac{y(x)}{x\sqrt{a^2-\frac{y(x{)}^2}{x^2}-1}}\right)}{2(a^2-1)}=c_1,y(x)],\text{Solve}[\frac{2i{\tan }^{-1}\left(\frac{y(x)}{x\sqrt{a^2-\frac{y(x{)}^2}{x^2}-1}}\right)+a(2\log (x-a^{2}x)-\log \left(\frac{(a^2-1)(y(x)-ix(a\sqrt{a^2-\frac{y(x{)}^2}{x^2}-1}+a^2-1))}{a^3(x-iy(x))}\right)+\log (-\frac{i(a^2-1)(x(a\sqrt{a^2-\frac{y(x{)}^2}{x^2}-1}+a^2-1)-iy(x))}{a^3(x+iy(x))})+\log (\frac{y(x{)}^2}{x^2}+1))}{2(a^2-1)}=c_1,y(x)]\}$$

Maple ✓
cpu = 0.032 (sec), leaf count = 55

$$\{[x\left(\mathit{\_}\mathit{T}\right)=\mathit{\_}\mathit{C}\mathit{1}\frac{1}{\sqrt{{\mathit{\_}\mathit{T}}^2+1}}{\left({\mathrm{e}}^{\frac{\mathit{A}\mathit{r}\mathit{c}\mathit{s}\mathit{i}\mathit{n}\mathit{h}\left(\mathit{\_}\mathit{T}\right)}{a}}\right)}^{-1},y\left(\mathit{\_}\mathit{T}\right)=\mathit{\_}\mathit{C}\mathit{1}(a\sqrt{{\mathit{\_}\mathit{T}}^2+1}+\mathit{\_}\mathit{T})\frac{1}{\sqrt{{\mathit{\_}\mathit{T}}^2+1}}{\left({\mathrm{e}}^{\frac{\mathit{A}\mathit{r}\mathit{c}\mathit{s}\mathit{i}\mathit{n}\mathit{h}\left(\mathit{\_}\mathit{T}\right)}{a}}\right)}^{-1}]\}$$ Mathematica raw input

DSolve[-y[x] + x*y'[x] + a*x*Sqrt[1 + y'[x]^2] == 0,y[x],x]

Mathematica raw output

{Solve[((-2*I)*ArcTan[y[x]/(x*Sqrt[-1 + a^2 - y[x]^2/x^2])] + a*(2*Log[x - a^2*x
] + Log[1 + y[x]^2/x^2] - Log[((-1 + a^2)*(y[x] + I*x*(-1 + a^2 + a*Sqrt[-1 + a^
2 - y[x]^2/x^2])))/(a^3*(x + I*y[x]))] + Log[(I*(-1 + a^2)*(I*y[x] + x*(-1 + a^2
 + a*Sqrt[-1 + a^2 - y[x]^2/x^2])))/(a^3*(x - I*y[x]))]))/(2*(-1 + a^2)) == C[1]
, y[x]], Solve[((2*I)*ArcTan[y[x]/(x*Sqrt[-1 + a^2 - y[x]^2/x^2])] + a*(2*Log[x 
- a^2*x] + Log[1 + y[x]^2/x^2] - Log[((-1 + a^2)*(y[x] - I*x*(-1 + a^2 + a*Sqrt[
-1 + a^2 - y[x]^2/x^2])))/(a^3*(x - I*y[x]))] + Log[((-I)*(-1 + a^2)*((-I)*y[x] 
+ x*(-1 + a^2 + a*Sqrt[-1 + a^2 - y[x]^2/x^2])))/(a^3*(x + I*y[x]))]))/(2*(-1 + 
a^2)) == C[1], y[x]]}

Maple raw input

dsolve(a*x*(1+diff(y(x),x)^2)^(1/2)+x*diff(y(x),x)-y(x) = 0, y(x),'implicit')

Maple raw output

[x(_T) = 1/(_T^2+1)^(1/2)/exp(1/a*arcsinh(_T))*_C1, y(_T) = (a*(_T^2+1)^(1/2)+_T
)/(_T^2+1)^(1/2)/exp(1/a*arcsinh(_T))*_C1]