ODE
\[ a x \sqrt {y'(x)^2+1}+x y'(x)-y(x)=0 \] ODE Classification
[[_homogeneous, `class A`], _dAlembert]
Book solution method
No Missing Variables ODE, Solve for \(y\)
Mathematica ✓
cpu = 0.789498 (sec), leaf count = 369
\[\left \{\text {Solve}\left [\frac {a \left (2 \log \left (x-a^2 x\right )-\log \left (\frac {\left (a^2-1\right ) \left (y(x)+i x \left (a \sqrt {a^2-\frac {y(x)^2}{x^2}-1}+a^2-1\right )\right )}{a^3 (x+i y(x))}\right )+\log \left (\frac {i \left (a^2-1\right ) \left (x \left (a \sqrt {a^2-\frac {y(x)^2}{x^2}-1}+a^2-1\right )+i y(x)\right )}{a^3 (x-i y(x))}\right )+\log \left (\frac {y(x)^2}{x^2}+1\right )\right )-2 i \tan ^{-1}\left (\frac {y(x)}{x \sqrt {a^2-\frac {y(x)^2}{x^2}-1}}\right )}{2 \left (a^2-1\right )}=c_1,y(x)\right ],\text {Solve}\left [\frac {2 i \tan ^{-1}\left (\frac {y(x)}{x \sqrt {a^2-\frac {y(x)^2}{x^2}-1}}\right )+a \left (2 \log \left (x-a^2 x\right )-\log \left (\frac {\left (a^2-1\right ) \left (y(x)-i x \left (a \sqrt {a^2-\frac {y(x)^2}{x^2}-1}+a^2-1\right )\right )}{a^3 (x-i y(x))}\right )+\log \left (-\frac {i \left (a^2-1\right ) \left (x \left (a \sqrt {a^2-\frac {y(x)^2}{x^2}-1}+a^2-1\right )-i y(x)\right )}{a^3 (x+i y(x))}\right )+\log \left (\frac {y(x)^2}{x^2}+1\right )\right )}{2 \left (a^2-1\right )}=c_1,y(x)\right ]\right \}\]
Maple ✓
cpu = 0.032 (sec), leaf count = 55
\[ \left \{ [x \left ( {\it \_T} \right ) ={{\it \_C1}{\frac {1}{\sqrt {{{\it \_T}}^{2}+1}}} \left ( {{\rm e}^{{\frac {{\it Arcsinh} \left ( {\it \_T} \right ) }{a}}}} \right ) ^{-1}},y \left ( {\it \_T} \right ) ={{\it \_C1} \left ( a\sqrt {{{\it \_T}}^{2}+1}+{\it \_T} \right ) {\frac {1}{\sqrt {{{\it \_T}}^{2}+1}}} \left ( {{\rm e}^{{\frac {{\it Arcsinh} \left ( {\it \_T} \right ) }{a}}}} \right ) ^{-1}}] \right \} \] 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]