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 = 0.232528 (sec), leaf count = 109
\[\left \{\left \{y(x)\to \frac {1}{2} \left (\frac {x \left (a x-\sqrt {a^2+x^2-1}\right )}{a^2-1}-\log \left (\sqrt {a^2+x^2-1}+x\right )\right )+c_1\right \},\left \{y(x)\to \frac {1}{2} \left (\frac {x \left (\sqrt {a^2+x^2-1}+a x\right )}{a^2-1}+\log \left (\sqrt {a^2+x^2-1}+x\right )\right )+c_1\right \}\right \}\]
Maple ✓
cpu = 0.166 (sec), leaf count = 195
\[\left [y \left (x \right ) = \frac {x \sqrt {a^{2}+x^{2}-1}}{2 \left (a -1\right ) \left (1+a \right )}+\frac {\ln \left (x +\sqrt {a^{2}+x^{2}-1}\right ) a^{2}}{2 \left (a -1\right ) \left (1+a \right )}-\frac {\ln \left (x +\sqrt {a^{2}+x^{2}-1}\right )}{2 \left (a -1\right ) \left (1+a \right )}+\frac {a \,x^{2}}{2 \left (a -1\right ) \left (1+a \right )}+\textit {\_C1}, y \left (x \right ) = -\frac {x \sqrt {a^{2}+x^{2}-1}}{2 \left (a -1\right ) \left (1+a \right )}-\frac {\ln \left (x +\sqrt {a^{2}+x^{2}-1}\right ) a^{2}}{2 \left (a -1\right ) \left (1+a \right )}+\frac {\ln \left (x +\sqrt {a^{2}+x^{2}-1}\right )}{2 \left (a -1\right ) \left (1+a \right )}+\frac {a \,x^{2}}{2 \left (a -1\right ) \left (1+a \right )}+\textit {\_C1}\right ]\] 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}, {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}}
Maple raw input
dsolve((1+diff(y(x),x)^2)^(1/2)+a*diff(y(x),x) = x, y(x))
Maple raw output
[y(x) = 1/2/(a-1)/(1+a)*x*(a^2+x^2-1)^(1/2)+1/2/(a-1)/(1+a)*ln(x+(a^2+x^2-1)^(1/2))*a^2-1/2/(a-1)/(1+a)*ln(x+(a^2+x^2-1)^(1/2))+1/2/(a-1)/(1+a)*a*x^2+_C1, y(x) = -1/2/(a-1)/(1+a)*x*(a^2+x^2-1)^(1/2)-1/2/(a-1)/(1+a)*ln(x+(a^2+x^2-1)^(1/2))*a^2+1/2/(a-1)/(1+a)*ln(x+(a^2+x^2-1)^(1/2))+1/2/(a-1)/(1+a)*a*x^2+_C1]