60.2.385 problem 962

Internal problem ID [10972]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 962
Date solved : Tuesday, January 28, 2025 at 05:40:00 PM
CAS classification : [_rational]

\begin{align*} 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 x^{2} y-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 x^{4} y^{2}+x^{6}} \end{align*}

✓ Solution by Maple

Time used: 0.125 (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)
 

\[ -\frac {y}{\left (a -1\right ) \left (a +1\right )}+\frac {2}{\left (a^{2}-1\right )^{2} \left (a^{2} x^{2}-x^{2}-y^{2}\right )^{2}}-\frac {2}{\left (a^{2}-1\right )^{2} \left (a^{2} x^{2}-x^{2}-y^{2}\right )}+c_{1} = 0 \]

✓ Solution by Mathematica

Time used: 3.615 (sec). Leaf size: 1743

DSolve[D[y[x],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]
 

\[ \text {Solve}\left [\int _1^x\left (\frac {4 \left (2 K[1]^3 a^8-8 K[1]^3 a^6-2 K[1] y(x)^2 a^6+2 K[1]^3 y(x) a^6+11 K[1]^3 a^4-2 K[1] y(x)^3 a^4+6 K[1] y(x)^2 a^4-6 K[1]^3 y(x) a^4-6 K[1]^3 a^2+4 K[1] y(x)^3 a^2-5 K[1] y(x)^2 a^2+2 K[1] a^2+6 K[1]^3 y(x) a^2+K[1]^3-2 K[1] y(x)^3+K[1] y(x)^2-2 K[1]-2 K[1]^3 y(x)\right )}{2 K[1]^4 a^8-8 K[1]^4 a^6-4 K[1]^2 y(x)^2 a^6+2 K[1]^4 y(x) a^6+11 K[1]^4 a^4+2 y(x)^4 a^4-4 K[1]^2 y(x)^3 a^4+12 K[1]^2 y(x)^2 a^4-6 K[1]^4 y(x) a^4+2 y(x)^5 a^2-6 K[1]^4 a^2-4 y(x)^4 a^2+8 K[1]^2 y(x)^3 a^2+4 K[1]^2 a^2-10 K[1]^2 y(x)^2 a^2+6 K[1]^4 y(x) a^2-2 y(x)^5+K[1]^4+y(x)^4-4 K[1]^2 y(x)^3-4 K[1]^2+2 K[1]^2 y(x)^2-4 y(x)^2-2 K[1]^4 y(x)-4}-\frac {4 \left (a^2-1\right ) K[1]}{a^2 K[1]^2-K[1]^2-y(x)^2}\right )dK[1]+\int _1^{y(x)}\left (-\frac {4 K[2]}{-a^2 x^2+x^2+K[2]^2}-\int _1^x\left (-\frac {8 \left (a^2-1\right ) K[1] K[2]}{\left (a^2 K[1]^2-K[1]^2-K[2]^2\right )^2}+\frac {4 \left (2 K[1]^3 a^6-4 K[1] K[2] a^6-6 K[1]^3 a^4-6 K[1] K[2]^2 a^4+12 K[1] K[2] a^4+6 K[1]^3 a^2+12 K[1] K[2]^2 a^2-10 K[1] K[2] a^2-2 K[1]^3-6 K[1] K[2]^2+2 K[1] K[2]\right )}{2 K[1]^4 a^8-8 K[1]^4 a^6-4 K[1]^2 K[2]^2 a^6+2 K[1]^4 K[2] a^6+11 K[1]^4 a^4+2 K[2]^4 a^4-4 K[1]^2 K[2]^3 a^4+12 K[1]^2 K[2]^2 a^4-6 K[1]^4 K[2] a^4+2 K[2]^5 a^2-6 K[1]^4 a^2-4 K[2]^4 a^2+8 K[1]^2 K[2]^3 a^2+4 K[1]^2 a^2-10 K[1]^2 K[2]^2 a^2+6 K[1]^4 K[2] a^2-2 K[2]^5+K[1]^4+K[2]^4-4 K[1]^2 K[2]^3-4 K[1]^2+2 K[1]^2 K[2]^2-4 K[2]^2-2 K[1]^4 K[2]-4}-\frac {4 \left (2 K[1]^3 a^8-8 K[1]^3 a^6-2 K[1] K[2]^2 a^6+2 K[1]^3 K[2] a^6+11 K[1]^3 a^4-2 K[1] K[2]^3 a^4+6 K[1] K[2]^2 a^4-6 K[1]^3 K[2] a^4-6 K[1]^3 a^2+4 K[1] K[2]^3 a^2-5 K[1] K[2]^2 a^2+2 K[1] a^2+6 K[1]^3 K[2] a^2+K[1]^3-2 K[1] K[2]^3+K[1] K[2]^2-2 K[1]-2 K[1]^3 K[2]\right ) \left (2 K[1]^4 a^6-8 K[1]^2 K[2] a^6-6 K[1]^4 a^4+8 K[2]^3 a^4-12 K[1]^2 K[2]^2 a^4+24 K[1]^2 K[2] a^4+6 K[1]^4 a^2+10 K[2]^4 a^2-16 K[2]^3 a^2+24 K[1]^2 K[2]^2 a^2-20 K[1]^2 K[2] a^2-2 K[1]^4-10 K[2]^4+4 K[2]^3-12 K[1]^2 K[2]^2+4 K[1]^2 K[2]-8 K[2]\right )}{\left (2 K[1]^4 a^8-8 K[1]^4 a^6-4 K[1]^2 K[2]^2 a^6+2 K[1]^4 K[2] a^6+11 K[1]^4 a^4+2 K[2]^4 a^4-4 K[1]^2 K[2]^3 a^4+12 K[1]^2 K[2]^2 a^4-6 K[1]^4 K[2] a^4+2 K[2]^5 a^2-6 K[1]^4 a^2-4 K[2]^4 a^2+8 K[1]^2 K[2]^3 a^2+4 K[1]^2 a^2-10 K[1]^2 K[2]^2 a^2+6 K[1]^4 K[2] a^2-2 K[2]^5+K[1]^4+K[2]^4-4 K[1]^2 K[2]^3-4 K[1]^2+2 K[1]^2 K[2]^2-4 K[2]^2-2 K[1]^4 K[2]-4\right )^2}\right )dK[1]+\frac {2 \left (x^4 a^6-4 x^2 K[2] a^6-3 x^4 a^4+4 K[2]^3 a^4-6 x^2 K[2]^2 a^4+12 x^2 K[2] a^4+3 x^4 a^2+5 K[2]^4 a^2-8 K[2]^3 a^2+12 x^2 K[2]^2 a^2-10 x^2 K[2] a^2-x^4-5 K[2]^4+2 K[2]^3-6 x^2 K[2]^2+2 x^2 K[2]-4 K[2]\right )}{2 x^4 a^8-8 x^4 a^6-4 x^2 K[2]^2 a^6+2 x^4 K[2] a^6+11 x^4 a^4+2 K[2]^4 a^4-4 x^2 K[2]^3 a^4+12 x^2 K[2]^2 a^4-6 x^4 K[2] a^4+2 K[2]^5 a^2-6 x^4 a^2-4 K[2]^4 a^2+8 x^2 K[2]^3 a^2+4 x^2 a^2-10 x^2 K[2]^2 a^2+6 x^4 K[2] a^2-2 K[2]^5+x^4+K[2]^4-4 x^2 K[2]^3-4 x^2+2 x^2 K[2]^2-4 K[2]^2-2 x^4 K[2]-4}\right )dK[2]=c_1,y(x)\right ] \]