Internal problem ID [9234]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 900.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_1st_order, _with_linear_symmetries], _rational]
\[ \boxed {y^{\prime }-\frac {2 a \left (-y^{2}+4 a x -1\right )}{-y^{3}+4 y a x -y-2 a y^{6}+24 y^{4} a^{2} x -96 y^{2} a^{3} x^{2}+128 a^{4} x^{3}}=0} \]
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 48
dsolve(diff(y(x),x) = 2*a*(-y(x)^2+4*a*x-1)/(-y(x)^3+4*y(x)*a*x-y(x)-2*a*y(x)^6+24*y(x)^4*a^2*x-96*y(x)^2*a^3*x^2+128*a^4*x^3),y(x), singsol=all)
\[ \frac {y \left (x \right )}{2 a}-\frac {1}{16 \left (-4 a x +y \left (x \right )^{2}\right )^{2} a^{2}}+\frac {1}{32 x \,a^{3}-8 a^{2} y \left (x \right )^{2}}-c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 60.135 (sec). Leaf size: 381
DSolve[y'[x] == (2*a*(-1 + 4*a*x - y[x]^2))/(128*a^4*x^3 - y[x] + 4*a*x*y[x] - 96*a^3*x^2*y[x]^2 - y[x]^3 + 24*a^2*x*y[x]^4 - 2*a*y[x]^6),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \text {Root}\left [8 \text {$\#$1}^5 a-16 \text {$\#$1}^4 a^2 c_1-64 \text {$\#$1}^3 a^2 x+\text {$\#$1}^2 \left (-2+128 a^3 c_1 x\right )+128 \text {$\#$1} a^3 x^2-256 a^4 c_1 x^2+8 a x-1\&,1\right ] \\ y(x)\to \text {Root}\left [8 \text {$\#$1}^5 a-16 \text {$\#$1}^4 a^2 c_1-64 \text {$\#$1}^3 a^2 x+\text {$\#$1}^2 \left (-2+128 a^3 c_1 x\right )+128 \text {$\#$1} a^3 x^2-256 a^4 c_1 x^2+8 a x-1\&,2\right ] \\ y(x)\to \text {Root}\left [8 \text {$\#$1}^5 a-16 \text {$\#$1}^4 a^2 c_1-64 \text {$\#$1}^3 a^2 x+\text {$\#$1}^2 \left (-2+128 a^3 c_1 x\right )+128 \text {$\#$1} a^3 x^2-256 a^4 c_1 x^2+8 a x-1\&,3\right ] \\ y(x)\to \text {Root}\left [8 \text {$\#$1}^5 a-16 \text {$\#$1}^4 a^2 c_1-64 \text {$\#$1}^3 a^2 x+\text {$\#$1}^2 \left (-2+128 a^3 c_1 x\right )+128 \text {$\#$1} a^3 x^2-256 a^4 c_1 x^2+8 a x-1\&,4\right ] \\ y(x)\to \text {Root}\left [8 \text {$\#$1}^5 a-16 \text {$\#$1}^4 a^2 c_1-64 \text {$\#$1}^3 a^2 x+\text {$\#$1}^2 \left (-2+128 a^3 c_1 x\right )+128 \text {$\#$1} a^3 x^2-256 a^4 c_1 x^2+8 a x-1\&,5\right ] \\ \end{align*}