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 = 1.21769 (sec), leaf count = 309
\[\left \{\text {Solve}\left [\frac {2 i \tan ^{-1}\left (\frac {y(x)}{x \sqrt {a^2-\frac {y(x)^2}{x^2}-1}}\right )+a \tanh ^{-1}\left (\frac {-a^2-\frac {i y(x)}{x}+1}{a \sqrt {a^2-\frac {y(x)^2}{x^2}-1}}\right )-a \tanh ^{-1}\left (\frac {-a^2+\frac {i y(x)}{x}+1}{a \sqrt {a^2-\frac {y(x)^2}{x^2}-1}}\right )+2 a \log \left (x-a^2 x\right )+a \log \left (\frac {y(x)^2}{x^2}+1\right )}{2 a^2-2}=c_1,y(x)\right ],\text {Solve}\left [\frac {-2 i \tan ^{-1}\left (\frac {y(x)}{x \sqrt {a^2-\frac {y(x)^2}{x^2}-1}}\right )-a \tanh ^{-1}\left (\frac {-a^2-\frac {i y(x)}{x}+1}{a \sqrt {a^2-\frac {y(x)^2}{x^2}-1}}\right )+a \tanh ^{-1}\left (\frac {-a^2+\frac {i y(x)}{x}+1}{a \sqrt {a^2-\frac {y(x)^2}{x^2}-1}}\right )+2 a \log \left (x-a^2 x\right )+a \log \left (\frac {y(x)^2}{x^2}+1\right )}{2 a^2-2}=c_1,y(x)\right ]\right \}\]
Maple ✓
cpu = 2.654 (sec), leaf count = 223
\[\left [x -\frac {{\mathrm e}^{\frac {\arcsinh \left (\frac {\sqrt {-a^{2} x^{2}+x^{2}+y \left (x \right )^{2}}\, a +y \left (x \right )}{\left (a^{2}-1\right ) x}\right )}{a}} \textit {\_C1}}{\sqrt {\frac {-a^{2} x^{2}+a^{2} y \left (x \right )^{2}+2 \sqrt {-a^{2} x^{2}+x^{2}+y \left (x \right )^{2}}\, a y \left (x \right )+x^{2}+y \left (x \right )^{2}}{\left (a^{2}-1\right )^{2} x^{2}}}} = 0, x -\frac {{\mathrm e}^{-\frac {\arcsinh \left (\frac {\sqrt {-a^{2} x^{2}+x^{2}+y \left (x \right )^{2}}\, a -y \left (x \right )}{\left (a^{2}-1\right ) x}\right )}{a}} \textit {\_C1}}{\sqrt {-\frac {a^{2} x^{2}-a^{2} y \left (x \right )^{2}+2 \sqrt {-a^{2} x^{2}+x^{2}+y \left (x \right )^{2}}\, a y \left (x \right )-x^{2}-y \left (x \right )^{2}}{\left (a^{2}-1\right )^{2} x^{2}}}} = 0\right ]\] 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*ArcTanh[(1 - a^2 - (I*y[x])/x)/(a*Sqrt[-1 + a^2 - y[x]^2/x^2])] - a*ArcTanh[(1 - a^2 + (I*y[x])/x)/(a*Sqrt[-1 + a^2 - y[x]^2/x^2])] + 2*a*Log[x - a^2*x] + a*Log[1 + y[x]^2/x^2])/(-2 + 2*a^2) == C[1], y[x]], Solve[((-2*I)*ArcTan[y[x]/(x*Sqrt[-1 + a^2 - y[x]^2/x^2])] - a*ArcTanh[(1 - a^2 - (I*y[x])/x)/(a*Sqrt[-1 + a^2 - y[x]^2/x^2])] + a*ArcTanh[(1 - a^2 + (I*y[x])/x)/(a*Sqrt[-1 + a^2 - y[x]^2/x^2])] + 2*a*Log[x - a^2*x] + a*Log[1 + y[x]^2/x^2])/(-2 + 2*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))
Maple raw output
[x-1/((-a^2*x^2+a^2*y(x)^2+2*(-a^2*x^2+x^2+y(x)^2)^(1/2)*a*y(x)+x^2+y(x)^2)/(a^2-1)^2/x^2)^(1/2)*exp(1/a*arcsinh(((-a^2*x^2+x^2+y(x)^2)^(1/2)*a+y(x))/(a^2-1)/x))*_C1 = 0, x-1/(-(a^2*x^2-a^2*y(x)^2+2*(-a^2*x^2+x^2+y(x)^2)^(1/2)*a*y(x)-x^2-y(x)^2)/(a^2-1)^2/x^2)^(1/2)*exp(-1/a*arcsinh(((-a^2*x^2+x^2+y(x)^2)^(1/2)*a-y(x))/(a^2-1)/x))*_C1 = 0]