60.2.50 problem 626

Internal problem ID [10637]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 626
Date solved : Tuesday, January 28, 2025 at 05:01:02 PM
CAS classification : [_rational, [_Abel, `2nd type`, `class C`]]

\begin{align*} y^{\prime }&=\frac {x}{y+\sqrt {x^{2}+1}} \end{align*}

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 118

dsolve(diff(y(x),x) = x/(y(x)+(x^2+1)^(1/2)),y(x), singsol=all)
 

\[ -\frac {2 \ln \left (11\right )}{3}+\frac {2 \ln \left (\frac {-y \sqrt {x^{2}+1}+x^{2}-y^{2}+1}{\left (y+\sqrt {x^{2}+1}\right )^{2}}\right )}{3}-\frac {4 \sqrt {5}\, \operatorname {arctanh}\left (\frac {\left (3 \sqrt {x^{2}+1}+y\right ) \sqrt {5}}{5 y+5 \sqrt {x^{2}+1}}\right )}{15}-\frac {4 \ln \left (\frac {\sqrt {x^{2}+1}}{y+\sqrt {x^{2}+1}}\right )}{3}+\frac {2 \ln \left (x^{2}+1\right )}{3}-c_{1} = 0 \]

✓ Solution by Mathematica

Time used: 0.160 (sec). Leaf size: 88

DSolve[D[y[x],x] == x/(Sqrt[1 + x^2] + y[x]),y[x],x,IncludeSingularSolutions -> True]
 

\[ \text {Solve}\left [\frac {1}{2} \left (\log \left (-\frac {y(x)^2}{x^2+1}-\frac {y(x)}{\sqrt {x^2+1}}+1\right )+\log \left (x^2+1\right )\right )=\frac {\text {arctanh}\left (\frac {3 \sqrt {x^2+1}+y(x)}{\sqrt {5} \left (\sqrt {x^2+1}+y(x)\right )}\right )}{\sqrt {5}}+c_1,y(x)\right ] \]