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 = 0.238594 (sec), leaf count = 273
\[\left \{\left \{y(x)\to \text {InverseFunction}\left [\frac {2 \log \left (\sqrt {\text {$\#$1}^2+a^2-1}+\text {$\#$1}\right )-a \log \left (-a \sqrt {\text {$\#$1}^2+a^2-1}-\text {$\#$1}-a^2+1\right )+a \log \left (-a \sqrt {\text {$\#$1}^2+a^2-1}+\text {$\#$1}-a^2+1\right )-a \log \left (1-\text {$\#$1}^2\right )+a \log (1-\text {$\#$1})-a \log (\text {$\#$1}+1)}{2-2 a^2}\& \right ]\left [\frac {x}{a^2-1}+c_1\right ]\right \},\left \{y(x)\to \text {InverseFunction}\left [\frac {2 \log \left (\sqrt {\text {$\#$1}^2+a^2-1}+\text {$\#$1}\right )-a \log \left (-a \sqrt {\text {$\#$1}^2+a^2-1}-\text {$\#$1}-a^2+1\right )+a \log \left (-a \sqrt {\text {$\#$1}^2+a^2-1}+\text {$\#$1}-a^2+1\right )+a \log \left (1-\text {$\#$1}^2\right )+a \log (1-\text {$\#$1})-a \log (\text {$\#$1}+1)}{2 \left (a^2-1\right )}\& \right ]\left [\frac {x}{a^2-1}+c_1\right ]\right \}\right \}\]
Maple ✓
cpu = 0.073 (sec), leaf count = 77
\[ \left \{ x-\int ^{y \left ( x \right ) }\!{ \left ( a-1 \right ) \left ( 1+a \right ) \left ( {\it \_a}\,a-\sqrt {{{\it \_a}}^{2}+{a}^{2}-1} \right ) ^{-1}}{d{\it \_a}}-{\it \_C1}=0,x-\int ^{y \left ( x \right ) }\!{ \left ( a-1 \right ) \left ( 1+a \right ) \left ( {\it \_a}\,a+\sqrt {{{\it \_a}}^{2}+{a}^{2}-1} \right ) ^{-1}}{d{\it \_a}}-{\it \_C1}=0 \right \} \] Mathematica raw input
DSolve[a*y'[x] + Sqrt[1 + y'[x]^2] == y[x],y[x],x]
Mathematica raw output
{{y[x] -> InverseFunction[(a*Log[1 - #1] - a*Log[1 + #1] - a*Log[1 - #1^2] + 2*Log[#1 + Sqrt[-1 + a^2 + #1^2]] - a*Log[1 - a^2 - #1 - a*Sqrt[-1 + a^2 + #1^2]] + a*Log[1 - a^2 + #1 - a*Sqrt[-1 + a^2 + #1^2]])/(2 - 2*a^2) & ][x/(-1 + a^2) + C[1]]}, {y[x] -> InverseFunction[(a*Log[1 - #1] - a*Log[1 + #1] + a*Log[1 - #1^2] + 2*Log[#1 + Sqrt[-1 + a^2 + #1^2]] - a*Log[1 - a^2 - #1 - a*Sqrt[-1 + a^2 + #1^2]] + a*Log[1 - a^2 + #1 - a*Sqrt[-1 + a^2 + #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),'implicit')
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