Internal problem ID [8494]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 914.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational]
Solve \begin {gather*} \boxed {y^{\prime }-\frac {2 a \left (x y^{2}-4 a +x \right )}{-x^{3} y^{3}+4 a \,x^{2} y-x^{3} y+2 a y^{6} x^{3}-24 y^{4} a^{2} x^{2}+96 y^{2} x \,a^{3}-128 a^{4}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.021 (sec). Leaf size: 77
dsolve(diff(y(x),x) = 2*a*(x*y(x)^2-4*a+x)/(-x^3*y(x)^3+4*a*x^2*y(x)-x^3*y(x)+2*a*y(x)^6*x^3-24*y(x)^4*a^2*x^2+96*y(x)^2*x*a^3-128*a^4),y(x), singsol=all)
\[ -\frac {-\frac {y \relax (x )^{2}+1}{y \relax (x )^{4} \left (x y \relax (x )^{2}-4 a \right )}-\frac {2 a}{y \relax (x )^{4} \left (x y \relax (x )^{2}-4 a \right )^{2}}}{2 a}+\frac {2 a y \relax (x )+\frac {1}{4 y \relax (x )^{4}}+\frac {1}{2 y \relax (x )^{2}}}{4 a^{2}}+c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 115.923 (sec). Leaf size: 401
DSolve[y'[x] == (2*a*(-4*a + x + x*y[x]^2))/(-128*a^4 + 4*a*x^2*y[x] - x^3*y[x] + 96*a^3*x*y[x]^2 - x^3*y[x]^3 - 24*a^2*x^2*y[x]^4 + 2*a*x^3*y[x]^6),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \text {Root}\left [8 \text {$\#$1}^5 a x^2-8 \text {$\#$1}^4 a c_1 x^2-64 \text {$\#$1}^3 a^2 x+\text {$\#$1}^2 \left (2 x^2+64 a^2 c_1 x\right )+128 \text {$\#$1} a^3-128 a^3 c_1-8 a x+x^2\&,1\right ] \\ y(x)\to \text {Root}\left [8 \text {$\#$1}^5 a x^2-8 \text {$\#$1}^4 a c_1 x^2-64 \text {$\#$1}^3 a^2 x+\text {$\#$1}^2 \left (2 x^2+64 a^2 c_1 x\right )+128 \text {$\#$1} a^3-128 a^3 c_1-8 a x+x^2\&,2\right ] \\ y(x)\to \text {Root}\left [8 \text {$\#$1}^5 a x^2-8 \text {$\#$1}^4 a c_1 x^2-64 \text {$\#$1}^3 a^2 x+\text {$\#$1}^2 \left (2 x^2+64 a^2 c_1 x\right )+128 \text {$\#$1} a^3-128 a^3 c_1-8 a x+x^2\&,3\right ] \\ y(x)\to \text {Root}\left [8 \text {$\#$1}^5 a x^2-8 \text {$\#$1}^4 a c_1 x^2-64 \text {$\#$1}^3 a^2 x+\text {$\#$1}^2 \left (2 x^2+64 a^2 c_1 x\right )+128 \text {$\#$1} a^3-128 a^3 c_1-8 a x+x^2\&,4\right ] \\ y(x)\to \text {Root}\left [8 \text {$\#$1}^5 a x^2-8 \text {$\#$1}^4 a c_1 x^2-64 \text {$\#$1}^3 a^2 x+\text {$\#$1}^2 \left (2 x^2+64 a^2 c_1 x\right )+128 \text {$\#$1} a^3-128 a^3 c_1-8 a x+x^2\&,5\right ] \\ \end{align*}