Internal problem ID [9114]
Internal file name [OUTPUT/8049_Monday_June_06_2022_01_23_01_AM_59189718/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 780.
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.
\[ \boxed {y^{\prime }-\frac {x y+y+\sqrt {y^{2}+x^{2}}\, x}{x \left (x +1\right )}=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & -x^{2} y^{\prime }+x y+\sqrt {y^{2}+x^{2}}\, x -y^{\prime } x +y=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-\frac {x y+y+\sqrt {y^{2}+x^{2}}\, x}{-x^{2}-x} \end {array} \]
Maple trace Kovacic algorithm successful
`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 -> Calling odsolve with the ODE`, diff(diff(y(x), x), x)+(-x^3*(diff(y(x), x))-3*x^2*(diff(y(x), x))+3*x*y(x)-2*x*(diff(y(x), x))+2* Methods for second order ODEs: --- Trying classification methods --- trying a quadrature checking if the LODE has constant coefficients checking if the LODE is of Euler type trying a symmetry of the form [xi=0, eta=F(x)] checking if the LODE is missing y -> Trying a Liouvillian solution using Kovacics algorithm A Liouvillian solution exists Reducible group (found an exponential solution) <- Kovacics algorithm successful <- 1st order ODE linearizable_by_differentiation successful`
✓ Solution by Maple
Time used: 0.406 (sec). Leaf size: 32
dsolve(diff(y(x),x) = (x*y(x)+y(x)+x*(y(x)^2+x^2)^(1/2))/x/(x+1),y(x), singsol=all)
\[ \frac {\sqrt {y \left (x \right )^{2}+x^{2}}+y \left (x \right )+\left (x^{2}+x \right ) c_{1}}{x \left (x +1\right )} = 0 \]
✓ Solution by Mathematica
Time used: 0.463 (sec). Leaf size: 35
DSolve[y'[x] == (y[x] + x*y[x] + x*Sqrt[x^2 + y[x]^2])/(x*(1 + x)),y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {e^{-c_1} x \left (-1+e^{2 c_1} (x+1)^2\right )}{2 (x+1)} \]