Internal problem ID [9235]
Internal file name [OUTPUT/8171_Monday_June_06_2022_02_01_57_AM_63520497/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 902.
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.
\[ \boxed {y^{\prime }-\frac {-y^{2} x +x^{3}-x -y^{6}+3 y^{4} x^{2}-3 y^{2} x^{4}+x^{6}}{\left (-y^{2}+x^{2}-1\right ) y}=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & -y^{6}+3 y^{4} x^{2}-3 y^{2} x^{4}+x^{6}+y^{\prime } y^{3}-y^{\prime } y x^{2}-y^{2} x +x^{3}+y^{\prime } y-x =0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {y^{6}-3 y^{4} x^{2}+3 y^{2} x^{4}-x^{6}+y^{2} x -x^{3}+x}{y^{3}-x^{2} y+y} \end {array} \]
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^6+3*x^4*y^2-3*x^2*y^4+y^6)/(x^2-y^2-1)/y], [0, (-1/4*x^2+1/4*y^2+1/2*x^
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 177
dsolve(diff(y(x),x) = (-x*y(x)^2+x^3-x-y(x)^6+3*x^2*y(x)^4-3*x^4*y(x)^2+x^6)/(-y(x)^2+x^2-1)/y(x),y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \frac {\sqrt {\left (-x +c_{1} \right ) \left (4 c_{1} x^{2}-4 x^{3}+\sqrt {4 c_{1} -4 x +1}+1\right )}}{2 x -2 c_{1}} \\ y \left (x \right ) &= \frac {\sqrt {\left (-x +c_{1} \right ) \left (4 c_{1} x^{2}-4 x^{3}+\sqrt {4 c_{1} -4 x +1}+1\right )}}{-2 x +2 c_{1}} \\ y \left (x \right ) &= \frac {\sqrt {\left (-4 c_{1} x^{2}+4 x^{3}+\sqrt {4 c_{1} -4 x +1}-1\right ) \left (x -c_{1} \right )}}{2 x -2 c_{1}} \\ y \left (x \right ) &= \frac {\sqrt {\left (-4 c_{1} x^{2}+4 x^{3}+\sqrt {4 c_{1} -4 x +1}-1\right ) \left (x -c_{1} \right )}}{-2 x +2 c_{1}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 8.232 (sec). Leaf size: 219
DSolve[y'[x] == (-x + x^3 + x^6 - x*y[x]^2 - 3*x^4*y[x]^2 + 3*x^2*y[x]^4 - y[x]^6)/(y[x]*(-1 + x^2 - y[x]^2)),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {1}{2} \sqrt {-\frac {-4 x^3+4 c_1 x^2+\sqrt {-4 x+1+4 c_1}+1}{x-c_1}} \\ y(x)\to \frac {1}{2} \sqrt {-\frac {-4 x^3+4 c_1 x^2+\sqrt {-4 x+1+4 c_1}+1}{x-c_1}} \\ y(x)\to -\frac {1}{2} \sqrt {\frac {4 x^3-4 c_1 x^2+\sqrt {-4 x+1+4 c_1}-1}{x-c_1}} \\ y(x)\to \frac {1}{2} \sqrt {\frac {4 x^3-4 c_1 x^2+\sqrt {-4 x+1+4 c_1}-1}{x-c_1}} \\ y(x)\to -\sqrt {x^2} \\ y(x)\to \sqrt {x^2} \\ \end{align*}