Internal problem ID [8542]
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.
CAS Maple gives this as type [_rational]
Solve \begin {gather*} \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 y a^{2} x^{2}-4 y x^{2}-8 y-y^{6} a^{2}+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 y^{4} x^{2}+3 x^{4} y^{2}+x^{6}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.062 (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)
\[ -\frac {y \relax (x )}{\left (a -1\right ) \left (a +1\right )}+\frac {2}{\left (a^{2}-1\right )^{2} \left (a^{2} x^{2}-x^{2}-y \relax (x )^{2}\right )^{2}}-\frac {2}{\left (a^{2}-1\right )^{2} \left (a^{2} x^{2}-x^{2}-y \relax (x )^{2}\right )}+c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 14.788 (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*}