Internal problem ID [7916]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 336.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_exact]
Solve \begin {gather*} \boxed {\left (\sqrt {y^{2}+1}+a x \right ) y^{\prime }+\sqrt {x^{2}+1}+a y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 41
dsolve(((y(x)^2+1)^(1/2)+a*x)*diff(y(x),x)+(x^2+1)^(1/2)+a*y(x) = 0,y(x), singsol=all)
\[ \frac {x \sqrt {x^{2}+1}}{2}+\frac {\arcsinh \relax (x )}{2}+a x y \relax (x )+\frac {y \relax (x ) \sqrt {1+y \relax (x )^{2}}}{2}+\frac {\arcsinh \left (y \relax (x )\right )}{2}+c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 0.792 (sec). Leaf size: 80
DSolve[Sqrt[1 + x^2] + a*y[x] + (a*x + Sqrt[1 + y[x]^2])*y'[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [a x y(x)+\frac {1}{2} \sqrt {x^2+1} x-\frac {1}{2} \log \left (\sqrt {x^2+1}-x\right )+\frac {1}{2} y(x) \sqrt {y(x)^2+1}-\frac {1}{2} \log \left (\sqrt {y(x)^2+1}-y(x)\right )=c_1,y(x)\right ] \]