Internal problem ID [7952]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 372.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [_quadrature]
Solve \begin {gather*} \boxed {\left (y^{\prime }\right )^{2}-4 y^{3}+a y+b=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 271
dsolve(diff(y(x),x)^2-4*y(x)^3+a*y(x)+b = 0,y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {\left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {1}{3}}}{6}+\frac {a}{2 \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {1}{3}}} \\ y \relax (x ) = -\frac {\left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {1}{3}}}{12}-\frac {a}{4 \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {1}{3}}}{6}-\frac {a}{2 \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {1}{3}}}\right )}{2} \\ y \relax (x ) = -\frac {\left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {1}{3}}}{12}-\frac {a}{4 \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {1}{3}}}{6}-\frac {a}{2 \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {1}{3}}}\right )}{2} \\ y \relax (x ) = \WeierstrassP \left (x +c_{1}, a , b\right ) \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.397 (sec). Leaf size: 244
DSolve[b + a*y[x] - 4*y[x]^3 + y'[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \wp (x-c_1;a,b) \\ y(x)\to \wp (x+c_1;a,b) \\ y(x)\to \frac {\left (\sqrt {81 b^2-3 a^3}+9 b\right )^{2/3}+\sqrt [3]{3} a}{2\ 3^{2/3} \sqrt [3]{\sqrt {81 b^2-3 a^3}+9 b}} \\ y(x)\to \frac {-2 \sqrt [3]{-3} a+i \left (\sqrt {3}+i\right ) \left (\sqrt {81 b^2-3 a^3}+9 b\right )^{2/3}}{4\ 3^{2/3} \sqrt [3]{\sqrt {81 b^2-3 a^3}+9 b}} \\ y(x)\to \frac {(-3)^{2/3} a-\sqrt [3]{-3} \left (\sqrt {81 b^2-3 a^3}+9 b\right )^{2/3}}{6 \sqrt [3]{\sqrt {81 b^2-3 a^3}+9 b}} \\ \end{align*}