Internal problem ID [9212]
Internal file name [OUTPUT/8148_Monday_June_06_2022_01_56_33_AM_41969472/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 879.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[[_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]
Unable to solve or complete the solution.
\[ \boxed {y^{\prime }+\frac {-x y-y+\sqrt {y^{2}+x^{2}}\, x^{2}-x y \sqrt {y^{2}+x^{2}}}{x \left (x +1\right )}=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x^{2} y^{\prime }-x y \sqrt {y^{2}+x^{2}}+\sqrt {y^{2}+x^{2}}\, x^{2}+y^{\prime } x -x y-y=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {x y \sqrt {y^{2}+x^{2}}-\sqrt {y^{2}+x^{2}}\, x^{2}+x y+y}{x^{2}+x} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying homogeneous types: differential order: 1; looking for linear symmetries trying exact Looking for potential symmetries trying an equivalence to an Abel ODE trying 1st order ODE linearizable_by_differentiation --- Trying Lie symmetry methods, 1st order --- `, `-> Computing symmetries using: way = 3 `, `-> Computing symmetries using: way = 5`[0, (x-y)*(x^2+y^2)^(1/2)/x]
✓ Solution by Maple
Time used: 0.062 (sec). Leaf size: 56
dsolve(diff(y(x),x) = -(-x*y(x)-y(x)+(y(x)^2+x^2)^(1/2)*x^2-x*(y(x)^2+x^2)^(1/2)*y(x))/x/(x+1),y(x), singsol=all)
\[ \ln \left (2\right )+\ln \left (\frac {x \left (\sqrt {2 y \left (x \right )^{2}+2 x^{2}}+y \left (x \right )+x \right )}{y \left (x \right )-x}\right )+\sqrt {2}\, x -\sqrt {2}\, \ln \left (x +1\right )-\ln \left (x \right )-c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 5.179 (sec). Leaf size: 81
DSolve[y'[x] == (y[x] + x*y[x] - x^2*Sqrt[x^2 + y[x]^2] + x*y[x]*Sqrt[x^2 + y[x]^2])/(x*(1 + x)),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {x \tanh \left (\frac {x-\log (x+1)+c_1}{\sqrt {2}}\right ) \left (2+\sqrt {2} \tanh \left (\frac {x-\log (x+1)+c_1}{\sqrt {2}}\right )\right )}{\sqrt {2}+2 \tanh \left (\frac {x-\log (x+1)+c_1}{\sqrt {2}}\right )} \\ y(x)\to x \\ \end{align*}