ODE
\[ a y'(x)+\sqrt {y'(x)^2+1}=x \] ODE Classification
[_quadrature]
Book solution method
Missing Variables ODE, Dependent variable missing, Solve for \(x\)
Mathematica ✓
cpu = 0.07209 (sec), leaf count = 120
\[\left \{\left \{y(x)\to \frac {x \left (a x-\sqrt {a^2+x^2-1}\right )-\left (a^2-1\right ) \log \left (\sqrt {a^2+x^2-1}+x\right )}{2 \left (a^2-1\right )}+c_1\right \},\left \{y(x)\to \frac {x \left (\sqrt {a^2+x^2-1}+a x\right )+\left (a^2-1\right ) \log \left (\sqrt {a^2+x^2-1}+x\right )}{2 \left (a^2-1\right )}+c_1\right \}\right \}\]
Maple ✓
cpu = 0.079 (sec), leaf count = 117
\[ \left \{ y \left ( x \right ) ={\frac {1}{ \left ( 2\,a-2 \right ) \left ( 1+a \right ) } \left ( a{x}^{2}-x\sqrt {{a}^{2}+{x}^{2}-1}- \left ( a-1 \right ) \left ( \ln \left ( x+\sqrt {{a}^{2}+{x}^{2}-1} \right ) -2\,{\it \_C1} \right ) \left ( 1+a \right ) \right ) },y \left ( x \right ) ={\frac {1}{ \left ( 2\,a-2 \right ) \left ( 1+a \right ) } \left ( a{x}^{2}+x\sqrt {{a}^{2}+{x}^{2}-1}+ \left ( a-1 \right ) \left ( \ln \left ( x+\sqrt {{a}^{2}+{x}^{2}-1} \right ) +2\,{\it \_C1} \right ) \left ( 1+a \right ) \right ) } \right \} \] Mathematica raw input
DSolve[a*y'[x] + Sqrt[1 + y'[x]^2] == x,y[x],x]
Mathematica raw output
{{y[x] -> C[1] + (x*(a*x - Sqrt[-1 + a^2 + x^2]) - (-1 + a^2)*Log[x + Sqrt[-1 + a^2 + x^2]])/(2*(-1 + a^2))}, {y[x] -> C[1] + (x*(a*x + Sqrt[-1 + a^2 + x^2]) + (-1 + a^2)*Log[x + Sqrt[-1 + a^2 + x^2]])/(2*(-1 + a^2))}}
Maple raw input
dsolve((1+diff(y(x),x)^2)^(1/2)+a*diff(y(x),x) = x, y(x),'implicit')
Maple raw output
y(x) = 1/2*(a*x^2+x*(a^2+x^2-1)^(1/2)+(a-1)*(ln(x+(a^2+x^2-1)^(1/2))+2*_C1)*(1+a))/(a-1)/(1+a), y(x) = 1/2*(a*x^2-x*(a^2+x^2-1)^(1/2)-(a-1)*(ln(x+(a^2+x^2-1)^(1/2))-2*_C1)*(1+a))/(a-1)/(1+a)