ODE
\[ a y'(x)+\sqrt {y'(x)^2+1}=y(x) \] ODE Classification
[_quadrature]
Book solution method
Missing Variables ODE, Independent variable missing, Solve for \(y\)
Mathematica ✓
cpu = 0.530081 (sec), leaf count = 241
\[\left \{\left \{y(x)\to \text {InverseFunction}\left [\frac {a \tanh ^{-1}\left (\frac {-\text {$\#$1}-a^2+1}{a \sqrt {\text {$\#$1}^2+a^2-1}}\right )+2 \tanh ^{-1}\left (\frac {\text {$\#$1}}{\sqrt {\text {$\#$1}^2+a^2-1}}\right )-a \tanh ^{-1}\left (\frac {\text {$\#$1}-a^2+1}{a \sqrt {\text {$\#$1}^2+a^2-1}}\right )+a \log \left (1-\text {$\#$1}^2\right )}{2 a^2-2}\& \right ]\left [\frac {x}{a^2-1}+c_1\right ]\right \},\left \{y(x)\to \text {InverseFunction}\left [\frac {-a \tanh ^{-1}\left (\frac {-\text {$\#$1}-a^2+1}{a \sqrt {\text {$\#$1}^2+a^2-1}}\right )-2 \tanh ^{-1}\left (\frac {\text {$\#$1}}{\sqrt {\text {$\#$1}^2+a^2-1}}\right )+a \left (\tanh ^{-1}\left (\frac {\text {$\#$1}-a^2+1}{a \sqrt {\text {$\#$1}^2+a^2-1}}\right )+\log \left (1-\text {$\#$1}^2\right )\right )}{2 \left (a^2-1\right )}\& \right ]\left [\frac {x}{a^2-1}+c_1\right ]\right \}\right \}\]
Maple ✓
cpu = 0.177 (sec), leaf count = 77
\[\left [x -\left (\int _{}^{y \left (x \right )}-\frac {\left (a -1\right ) \left (1+a \right )}{-a \textit {\_a} +\sqrt {\textit {\_a}^{2}+a^{2}-1}}d \textit {\_a} \right )-\textit {\_C1} = 0, x -\left (\int _{}^{y \left (x \right )}\frac {\left (a -1\right ) \left (1+a \right )}{a \textit {\_a} +\sqrt {\textit {\_a}^{2}+a^{2}-1}}d \textit {\_a} \right )-\textit {\_C1} = 0\right ]\] Mathematica raw input
DSolve[a*y'[x] + Sqrt[1 + y'[x]^2] == y[x],y[x],x]
Mathematica raw output
{{y[x] -> InverseFunction[(a*ArcTanh[(1 - a^2 - #1)/(a*Sqrt[-1 + a^2 + #1^2])] + 2*ArcTanh[#1/Sqrt[-1 + a^2 + #1^2]] - a*ArcTanh[(1 - a^2 + #1)/(a*Sqrt[-1 + a^2 + #1^2])] + a*Log[1 - #1^2])/(-2 + 2*a^2) & ][x/(-1 + a^2) + C[1]]}, {y[x] -> InverseFunction[(-(a*ArcTanh[(1 - a^2 - #1)/(a*Sqrt[-1 + a^2 + #1^2])]) - 2*ArcTanh[#1/Sqrt[-1 + a^2 + #1^2]] + a*(ArcTanh[(1 - a^2 + #1)/(a*Sqrt[-1 + a^2 + #1^2])] + Log[1 - #1^2]))/(2*(-1 + a^2)) & ][x/(-1 + a^2) + C[1]]}}
Maple raw input
dsolve((1+diff(y(x),x)^2)^(1/2)+a*diff(y(x),x) = y(x), y(x))
Maple raw output
[x-Intat(-1/(-a*_a+(_a^2+a^2-1)^(1/2))*(a-1)*(1+a),_a = y(x))-_C1 = 0, x-Intat(1/(a*_a+(_a^2+a^2-1)^(1/2))*(a-1)*(1+a),_a = y(x))-_C1 = 0]