Internal problem ID [9239]
Internal file name [OUTPUT/8175_Monday_June_06_2022_02_02_48_AM_56362697/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 906.
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.
\[ \boxed {y^{\prime }-\frac {x \left (x^{2}+y^{2}+1\right )}{-y^{3}-x^{2} y-y+y^{6}+3 y^{4} x^{2}+3 y^{2} x^{4}+x^{6}}=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime } y^{6}+3 y^{\prime } y^{4} x^{2}+3 y^{\prime } y^{2} x^{4}+y^{\prime } x^{6}-y^{\prime } y^{3}-y^{\prime } y x^{2}-y^{2} x -x^{3}-y^{\prime } y-x =0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {y^{2} x +x^{3}+x}{-y^{3}-x^{2} y-y+y^{6}+3 y^{4} x^{2}+3 y^{2} x^{4}+x^{6}} \end {array} \]
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 `, `-> Computing symmetries using: way = 3 `, `-> Computing symmetries using: way = 4`[-y/x, 1]
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 37
dsolve(diff(y(x),x) = x*(x^2+y(x)^2+1)/(-y(x)^3-x^2*y(x)-y(x)+y(x)^6+3*x^2*y(x)^4+3*x^4*y(x)^2+x^6),y(x), singsol=all)
\[ -\frac {1}{4 \left (y \left (x \right )^{2}+x^{2}\right )^{2}}-\frac {1}{2 y \left (x \right )^{2}+2 x^{2}}-y \left (x \right )+c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 60.098 (sec). Leaf size: 326
DSolve[y'[x] == (x*(1 + x^2 + y[x]^2))/(x^6 - y[x] - x^2*y[x] + 3*x^4*y[x]^2 - y[x]^3 + 3*x^2*y[x]^4 + y[x]^6),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \text {Root}\left [4 \text {$\#$1}^5-4 \text {$\#$1}^4 c_1+8 \text {$\#$1}^3 x^2+\text {$\#$1}^2 \left (2-8 c_1 x^2\right )+4 \text {$\#$1} x^4-4 c_1 x^4+2 x^2+1\&,1\right ] \\ y(x)\to \text {Root}\left [4 \text {$\#$1}^5-4 \text {$\#$1}^4 c_1+8 \text {$\#$1}^3 x^2+\text {$\#$1}^2 \left (2-8 c_1 x^2\right )+4 \text {$\#$1} x^4-4 c_1 x^4+2 x^2+1\&,2\right ] \\ y(x)\to \text {Root}\left [4 \text {$\#$1}^5-4 \text {$\#$1}^4 c_1+8 \text {$\#$1}^3 x^2+\text {$\#$1}^2 \left (2-8 c_1 x^2\right )+4 \text {$\#$1} x^4-4 c_1 x^4+2 x^2+1\&,3\right ] \\ y(x)\to \text {Root}\left [4 \text {$\#$1}^5-4 \text {$\#$1}^4 c_1+8 \text {$\#$1}^3 x^2+\text {$\#$1}^2 \left (2-8 c_1 x^2\right )+4 \text {$\#$1} x^4-4 c_1 x^4+2 x^2+1\&,4\right ] \\ y(x)\to \text {Root}\left [4 \text {$\#$1}^5-4 \text {$\#$1}^4 c_1+8 \text {$\#$1}^3 x^2+\text {$\#$1}^2 \left (2-8 c_1 x^2\right )+4 \text {$\#$1} x^4-4 c_1 x^4+2 x^2+1\&,5\right ] \\ \end{align*}