Internal problem ID [8615]
Internal file name [OUTPUT/7548_Sunday_June_05_2022_11_04_37_PM_99656506/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 279.
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]
Unable to solve or complete the solution.
\[ \boxed {\left (y^{2}+2 y+x \right ) y^{\prime }+\left (y+x \right )^{2} y^{2}+y \left (y+1\right )=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left (y^{2}+2 y+x \right ) y^{\prime }+\left (y+x \right )^{2} y^{2}+y \left (y+1\right )=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-\frac {\left (y+x \right )^{2} y^{2}+y \left (y+1\right )}{y^{2}+2 y+x} \end {array} \]
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 Looking for potential symmetries trying inverse_Riccati trying an equivalence to an Abel ODE differential order: 1; trying a linearization to 2nd order --- trying a change of variables {x -> y(x), y(x) -> x} differential order: 1; trying a linearization to 2nd order trying 1st order ODE linearizable_by_differentiation --- Trying Lie symmetry methods, 1st order --- `, `-> Computing symmetries using: way = 2`[0, (x^2*y^2+2*x*y^3+y^4)/(y^2+x+2*y)], [0, (x^3*y^2+2*x^2*y^3+x*y^4-x*y^2-y^3-
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 110
dsolve((y(x)^2+2*y(x)+x)*diff(y(x),x)+(y(x)+x)^2*y(x)^2+y(x)*(y(x)+1)=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \frac {x^{2}-c_{1} x +\sqrt {x^{4}-2 c_{1} x^{3}+\left (c_{1}^{2}-2\right ) x^{2}+\left (4+2 c_{1} \right ) x -4 c_{1} +1}-1}{2 c_{1} -2 x} \\ y \left (x \right ) &= \frac {-x^{2}+c_{1} x +\sqrt {x^{4}-2 c_{1} x^{3}+\left (c_{1}^{2}-2\right ) x^{2}+\left (4+2 c_{1} \right ) x -4 c_{1} +1}+1}{2 x -2 c_{1}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 2.179 (sec). Leaf size: 146
DSolve[(y[x]^2+2*y[x]+x)*y'[x]+(y[x]+x)^2*y[x]^2+y[x]*(y[x]+1)==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {x^2+\sqrt {\left (-x^2+c_1 x+1\right ){}^2+4 (x-c_1)}-c_1 x-1}{2 (x-c_1)} \\ y(x)\to \frac {-x^2+\sqrt {\left (-x^2+c_1 x+1\right ){}^2+4 (x-c_1)}+c_1 x+1}{2 (x-c_1)} \\ y(x)\to \frac {1}{2} \left (-\sqrt {x^2}-x\right ) \\ y(x)\to \frac {1}{2} \left (\sqrt {x^2}-x\right ) \\ \end{align*}