4.23.30 $-ay(x)y^{\prime }(x)-ax+\sqrt{y^{\prime }(x{)}^2+1}=0$

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

[[_1st_order, _with_linear_symmetries], _dAlembert]

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

Mathematica ✓
cpu = 96.7453 (sec), leaf count = 407

$$\{\text{Solve}[\sqrt{a^2(x^2+y(x{)}^2)-1}+\frac{1}{4}i\log ((a^2{x}^2-1)y(x{)}^2-2xy(x)\sqrt{a^2{x}^2+a^{2}y(x{)}^2-1}+a^{2}y(x{)}^4+x^2)+\frac{1}{4}i\log ((a^2{x}^2-1)y(x{)}^2+2xy(x)\sqrt{a^2{x}^2+a^{2}y(x{)}^2-1}+a^{2}y(x{)}^4+x^2)+{\tan }^{-1}\left(\frac{1}{\sqrt{a^2(x^2+y(x{)}^2)-1}}\right)-\frac{1}{2}i\log (1-a^{2}y(x{)}^2)-\frac{1}{2}i\log (x^2+y(x{)}^2)+{\tan }^{-1}\left(\frac{x}{y(x)}\right)=c_1,y(x)],\text{Solve}[4\sqrt{a^2{x}^2+a^{2}y(x{)}^2-1}+i\log (a^2{x}^{2}y(x{)}^2-2xy(x)\sqrt{a^2(x^2+y(x{)}^2)-1}+a^{2}y(x{)}^4+x^2-y(x{)}^2)+i\log (a^2{x}^{2}y(x{)}^2+2xy(x)\sqrt{a^2(x^2+y(x{)}^2)-1}+a^{2}y(x{)}^4+x^2-y(x{)}^2)+4{\tan }^{-1}\left(\frac{1}{\sqrt{a^2(x^2+y(x{)}^2)-1}}\right)-2i\log (a^{2}y(x{)}^2-1)+4c_1-2i\log (x^2+y(x{)}^2)=4{\tan }^{-1}\left(\frac{x}{y(x)}\right),y(x)]\}$$

Maple ✓
cpu = 0.043 (sec), leaf count = 53

$$\{[x\left(\mathit{\_}\mathit{T}\right)=\mathit{\_}\mathit{T}(\frac{\arctan \left(\mathit{\_}\mathit{T}\right)}{a}+\frac{1}{\mathit{\_}\mathit{T}a}+\mathit{\_}\mathit{C}\mathit{1})\frac{1}{\sqrt{{\mathit{\_}\mathit{T}}^2+1}},y\left(\mathit{\_}\mathit{T}\right)=-\frac{\mathit{\_}\mathit{C}\mathit{1}a-\mathit{\_}\mathit{T}+\arctan \left(\mathit{\_}\mathit{T}\right)}{a}\frac{1}{\sqrt{{\mathit{\_}\mathit{T}}^2+1}}]\}$$ Mathematica raw input

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

Mathematica raw output

{Solve[ArcTan[x/y[x]] + ArcTan[1/Sqrt[-1 + a^2*(x^2 + y[x]^2)]] - (I/2)*Log[x^2 
+ y[x]^2] - (I/2)*Log[1 - a^2*y[x]^2] + (I/4)*Log[x^2 + (-1 + a^2*x^2)*y[x]^2 + 
a^2*y[x]^4 - 2*x*y[x]*Sqrt[-1 + a^2*x^2 + a^2*y[x]^2]] + (I/4)*Log[x^2 + (-1 + a
^2*x^2)*y[x]^2 + a^2*y[x]^4 + 2*x*y[x]*Sqrt[-1 + a^2*x^2 + a^2*y[x]^2]] + Sqrt[-
1 + a^2*(x^2 + y[x]^2)] == C[1], y[x]], Solve[4*ArcTan[1/Sqrt[-1 + a^2*(x^2 + y[
x]^2)]] + 4*C[1] - (2*I)*Log[x^2 + y[x]^2] - (2*I)*Log[-1 + a^2*y[x]^2] + I*Log[
x^2 - y[x]^2 + a^2*x^2*y[x]^2 + a^2*y[x]^4 - 2*x*y[x]*Sqrt[-1 + a^2*(x^2 + y[x]^
2)]] + I*Log[x^2 - y[x]^2 + a^2*x^2*y[x]^2 + a^2*y[x]^4 + 2*x*y[x]*Sqrt[-1 + a^2
*(x^2 + y[x]^2)]] + 4*Sqrt[-1 + a^2*x^2 + a^2*y[x]^2] == 4*ArcTan[x/y[x]], y[x]]
}

Maple raw input

dsolve((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) = 1/(_T^2+1)^(1/2)*_T*(1/a*arctan(_T)+1/_T/a+_C1), y(_T) = -(_C1*a-_T+arc
tan(_T))/(_T^2+1)^(1/2)/a]