Internal problem ID [9295]
Internal file name [OUTPUT/8231_Monday_June_06_2022_02_24_25_AM_9875621/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 962.
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.
\[ \boxed {y^{\prime }-\frac {4 x \left (a -1\right ) \left (a +1\right ) \left (-y^{2}+a^{2} x^{2}-x^{2}-2\right )}{-4 y^{3}+4 a^{2} x^{2} y-4 y x^{2}-8 y-a^{2} y^{6}+3 a^{4} y^{4} x^{2}-6 y^{4} a^{2} x^{2}-3 a^{6} y^{2} x^{4}+9 y^{2} a^{4} x^{4}-9 y^{2} a^{2} x^{4}+a^{8} x^{6}-4 a^{6} x^{6}+6 a^{4} x^{6}-4 a^{2} x^{6}+y^{6}+3 x^{2} y^{4}+3 y^{2} x^{4}+x^{6}}=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & 4 y^{2} x -8 a^{2} x +4 a^{4} x^{3}-8 a^{2} x^{3}+4 y^{\prime } y x^{2}-6 y^{\prime } a^{4} x^{6}+4 y^{\prime } a^{2} x^{6}-3 y^{\prime } y^{4} x^{2}+4 y^{\prime } a^{6} x^{6}-y^{\prime } a^{8} x^{6}+y^{\prime } y^{6} a^{2}-3 y^{\prime } y^{2} x^{4}+4 x^{3}+4 y^{\prime } y^{3}-y^{\prime } y^{6}-y^{\prime } x^{6}+8 y^{\prime } y+3 y^{\prime } y^{2} a^{6} x^{4}-3 y^{\prime } y^{4} a^{4} x^{2}-9 y^{\prime } y^{2} a^{4} x^{4}+6 y^{\prime } y^{4} a^{2} x^{2}+9 y^{\prime } y^{2} a^{2} x^{4}-4 y^{\prime } y a^{2} x^{2}+8 x -4 y^{2} a^{2} x =0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {-4 y^{2} x +8 a^{2} x -4 a^{4} x^{3}+8 a^{2} x^{3}-4 x^{3}-8 x +4 y^{2} a^{2} x}{-a^{8} x^{6}+3 a^{6} y^{2} x^{4}+4 a^{6} x^{6}-3 a^{4} y^{4} x^{2}-9 y^{2} a^{4} x^{4}-6 a^{4} x^{6}+a^{2} y^{6}+6 y^{4} a^{2} x^{2}+9 y^{2} a^{2} x^{4}+4 a^{2} x^{6}-y^{6}-3 x^{2} y^{4}-3 y^{2} x^{4}-x^{6}-4 a^{2} x^{2} y+4 y^{3}+4 y x^{2}+8 y} \end {array} \]
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 `, `-> Computing symmetries using: way = 3 `, `-> Computing symmetries using: way = 4`[y/x, a^2-1]
✓ Solution by Maple
Time used: 0.063 (sec). Leaf size: 79
dsolve(diff(y(x),x) = 4*x*(a-1)*(a+1)*(-y(x)^2+a^2*x^2-x^2-2)/(-4*y(x)^3+4*a^2*x^2*y(x)-4*x^2*y(x)-8*y(x)-a^2*y(x)^6+3*a^4*y(x)^4*x^2-6*y(x)^4*a^2*x^2-3*a^6*y(x)^2*x^4+9*y(x)^2*a^4*x^4-9*y(x)^2*a^2*x^4+a^8*x^6-4*a^6*x^6+6*a^4*x^6-4*a^2*x^6+y(x)^6+3*x^2*y(x)^4+3*x^4*y(x)^2+x^6),y(x), singsol=all)
\[ -\frac {y \left (x \right )}{\left (a -1\right ) \left (a +1\right )}+\frac {2}{\left (a^{2}-1\right )^{2} \left (a^{2} x^{2}-x^{2}-y \left (x \right )^{2}\right )^{2}}-\frac {2}{\left (a^{2}-1\right )^{2} \left (a^{2} x^{2}-x^{2}-y \left (x \right )^{2}\right )}+c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 15.969 (sec). Leaf size: 1191
DSolve[y'[x] == (4*(-1 + a)*(1 + a)*x*(-2 - x^2 + a^2*x^2 - y[x]^2))/(x^6 - 4*a^2*x^6 + 6*a^4*x^6 - 4*a^6*x^6 + a^8*x^6 - 8*y[x] - 4*x^2*y[x] + 4*a^2*x^2*y[x] + 3*x^4*y[x]^2 - 9*a^2*x^4*y[x]^2 + 9*a^4*x^4*y[x]^2 - 3*a^6*x^4*y[x]^2 - 4*y[x]^3 + 3*x^2*y[x]^4 - 6*a^2*x^2*y[x]^4 + 3*a^4*x^2*y[x]^4 + y[x]^6 - a^2*y[x]^6),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \text {Root}\left [\text {$\#$1}^5 \left (2 a^2-2\right )+\text {$\#$1}^4 \left (2 a^4-4 a^2+1+e^{c_1}\right )+\text {$\#$1}^3 \left (-4 a^4 x^2+8 a^2 x^2-4 x^2\right )+\text {$\#$1}^2 \left (-4 a^6 x^2+12 a^4 x^2-10 a^2 x^2-2 a^2 e^{c_1} x^2+2 x^2+2 e^{c_1} x^2-4\right )+\text {$\#$1} \left (2 a^6 x^4-6 a^4 x^4+6 a^2 x^4-2 x^4\right )+2 a^8 x^4-8 a^6 x^4+11 a^4 x^4+a^4 e^{c_1} x^4-6 a^2 x^4-2 a^2 e^{c_1} x^4+4 a^2 x^2+x^4+e^{c_1} x^4-4 x^2-4\&,1\right ] \\ y(x)\to \text {Root}\left [\text {$\#$1}^5 \left (2 a^2-2\right )+\text {$\#$1}^4 \left (2 a^4-4 a^2+1+e^{c_1}\right )+\text {$\#$1}^3 \left (-4 a^4 x^2+8 a^2 x^2-4 x^2\right )+\text {$\#$1}^2 \left (-4 a^6 x^2+12 a^4 x^2-10 a^2 x^2-2 a^2 e^{c_1} x^2+2 x^2+2 e^{c_1} x^2-4\right )+\text {$\#$1} \left (2 a^6 x^4-6 a^4 x^4+6 a^2 x^4-2 x^4\right )+2 a^8 x^4-8 a^6 x^4+11 a^4 x^4+a^4 e^{c_1} x^4-6 a^2 x^4-2 a^2 e^{c_1} x^4+4 a^2 x^2+x^4+e^{c_1} x^4-4 x^2-4\&,2\right ] \\ y(x)\to \text {Root}\left [\text {$\#$1}^5 \left (2 a^2-2\right )+\text {$\#$1}^4 \left (2 a^4-4 a^2+1+e^{c_1}\right )+\text {$\#$1}^3 \left (-4 a^4 x^2+8 a^2 x^2-4 x^2\right )+\text {$\#$1}^2 \left (-4 a^6 x^2+12 a^4 x^2-10 a^2 x^2-2 a^2 e^{c_1} x^2+2 x^2+2 e^{c_1} x^2-4\right )+\text {$\#$1} \left (2 a^6 x^4-6 a^4 x^4+6 a^2 x^4-2 x^4\right )+2 a^8 x^4-8 a^6 x^4+11 a^4 x^4+a^4 e^{c_1} x^4-6 a^2 x^4-2 a^2 e^{c_1} x^4+4 a^2 x^2+x^4+e^{c_1} x^4-4 x^2-4\&,3\right ] \\ y(x)\to \text {Root}\left [\text {$\#$1}^5 \left (2 a^2-2\right )+\text {$\#$1}^4 \left (2 a^4-4 a^2+1+e^{c_1}\right )+\text {$\#$1}^3 \left (-4 a^4 x^2+8 a^2 x^2-4 x^2\right )+\text {$\#$1}^2 \left (-4 a^6 x^2+12 a^4 x^2-10 a^2 x^2-2 a^2 e^{c_1} x^2+2 x^2+2 e^{c_1} x^2-4\right )+\text {$\#$1} \left (2 a^6 x^4-6 a^4 x^4+6 a^2 x^4-2 x^4\right )+2 a^8 x^4-8 a^6 x^4+11 a^4 x^4+a^4 e^{c_1} x^4-6 a^2 x^4-2 a^2 e^{c_1} x^4+4 a^2 x^2+x^4+e^{c_1} x^4-4 x^2-4\&,4\right ] \\ y(x)\to \text {Root}\left [\text {$\#$1}^5 \left (2 a^2-2\right )+\text {$\#$1}^4 \left (2 a^4-4 a^2+1+e^{c_1}\right )+\text {$\#$1}^3 \left (-4 a^4 x^2+8 a^2 x^2-4 x^2\right )+\text {$\#$1}^2 \left (-4 a^6 x^2+12 a^4 x^2-10 a^2 x^2-2 a^2 e^{c_1} x^2+2 x^2+2 e^{c_1} x^2-4\right )+\text {$\#$1} \left (2 a^6 x^4-6 a^4 x^4+6 a^2 x^4-2 x^4\right )+2 a^8 x^4-8 a^6 x^4+11 a^4 x^4+a^4 e^{c_1} x^4-6 a^2 x^4-2 a^2 e^{c_1} x^4+4 a^2 x^2+x^4+e^{c_1} x^4-4 x^2-4\&,5\right ] \\ \end{align*}