Internal problem ID [8960]
Internal file name [OUTPUT/7895_Monday_June_06_2022_12_51_50_AM_69462615/index.tex]
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.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[_rational, [_Abel, `2nd type`, `class C`]]
Unable to solve or complete the solution.
\[ \boxed {y^{\prime }-\frac {x}{y+\sqrt {x^{2}+1}}=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y y^{\prime }+y^{\prime } \sqrt {x^{2}+1}-x =0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {x}{y+\sqrt {x^{2}+1}} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear trying Bernoulli trying separable trying inverse linear trying homogeneous types: trying Chini differential order: 1; looking for linear symmetries trying exact trying Abel <- Abel successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 118
dsolve(diff(y(x),x) = x/(y(x)+(x^2+1)^(1/2)),y(x), singsol=all)
\[ -\frac {2 \ln \left (11\right )}{3}+\frac {2 \ln \left (\frac {-y \left (x \right ) \sqrt {x^{2}+1}+x^{2}-y \left (x \right )^{2}+1}{\left (y \left (x \right )+\sqrt {x^{2}+1}\right )^{2}}\right )}{3}-\frac {4 \sqrt {5}\, \operatorname {arctanh}\left (\frac {\left (3 \sqrt {x^{2}+1}+y \left (x \right )\right ) \sqrt {5}}{5 y \left (x \right )+5 \sqrt {x^{2}+1}}\right )}{15}-\frac {4 \ln \left (\frac {\sqrt {x^{2}+1}}{y \left (x \right )+\sqrt {x^{2}+1}}\right )}{3}+\frac {2 \ln \left (x^{2}+1\right )}{3}-c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 0.16 (sec). Leaf size: 88
DSolve[y'[x] == x/(Sqrt[1 + x^2] + y[x]),y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\frac {1}{2} \left (\log \left (-\frac {y(x)^2}{x^2+1}-\frac {y(x)}{\sqrt {x^2+1}}+1\right )+\log \left (x^2+1\right )\right )=\frac {\text {arctanh}\left (\frac {3 \sqrt {x^2+1}+y(x)}{\sqrt {5} \left (\sqrt {x^2+1}+y(x)\right )}\right )}{\sqrt {5}}+c_1,y(x)\right ] \]