Internal problem ID [8204]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 626.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, [_Abel, `2nd type`, `class C`]]
\[ \boxed {y^{\prime }-\frac {x}{y+\sqrt {x^{2}+1}}=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 112
dsolve(diff(y(x),x) = x/(y(x)+(x^2+1)^(1/2)),y(x), singsol=all)
\[ -\frac {4 \ln \left (\frac {36 \sqrt {x^{2}+1}}{y \left (x \right )+\sqrt {x^{2}+1}}\right )}{3}+\frac {2 \ln \left (-\frac {1296 \left (\sqrt {x^{2}+1}\, y \left (x \right )-x^{2}+y \left (x \right )^{2}-1\right )}{11 \left (y \left (x \right )+\sqrt {x^{2}+1}\right )^{2}}\right )}{3}-\frac {4 \sqrt {5}\, \operatorname {arctanh}\left (\frac {\left (3 \sqrt {x^{2}+1}+y \left (x \right )\right ) \sqrt {5}}{5 y \left (x \right )+5 \sqrt {x^{2}+1}}\right )}{15}+\frac {2 \ln \left (x^{2}+1\right )}{3}-c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 0.168 (sec). Leaf size: 88
DSolve[y'[x] == x/(Sqrt[1 + x^2] + y[x]),y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\frac {1}{2} \left (\log \left (-\frac {y(x)^2}{x^2+1}-\frac {y(x)}{\sqrt {x^2+1}}+1\right )+\log \left (x^2+1\right )\right )=\frac {\text {arctanh}\left (\frac {3 \sqrt {x^2+1}+y(x)}{\sqrt {5} \left (\sqrt {x^2+1}+y(x)\right )}\right )}{\sqrt {5}}+c_1,y(x)\right ] \]