Internal problem ID [9383]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1805 (book 6.214).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]
Solve \begin {gather*} \boxed {\left (4 y^{3}-a y-b \right ) y^{\prime \prime }-\left (6 y^{2}-\frac {a}{2}\right ) \left (y^{\prime }\right )^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.125 (sec). Leaf size: 292
dsolve((4*y(x)^3-a*y(x)-b)*diff(diff(y(x),x),x)-(6*y(x)^2-1/2*a)*diff(y(x),x)^2=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} \\ \int _{}^{y \relax (x )}\frac {1}{\sqrt {4 \textit {\_a}^{3}-\textit {\_a} a -b}}d \textit {\_a} -c_{1} x -c_{2} = 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 11.8 (sec). Leaf size: 415
DSolve[(a/2 - 6*y[x]^2)*y'[x]^2 + (-b - a*y[x] + 4*y[x]^3)*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\frac {2 \sqrt {\frac {y(x)-\text {Root}\left [4 \text {$\#$1}^3-\text {$\#$1} a-b\&,1\right ]}{\text {Root}\left [4 \text {$\#$1}^3-\text {$\#$1} a-b\&,3\right ]-\text {Root}\left [4 \text {$\#$1}^3-\text {$\#$1} a-b\&,1\right ]}} \sqrt {\frac {y(x)-\text {Root}\left [4 \text {$\#$1}^3-\text {$\#$1} a-b\&,2\right ]}{\text {Root}\left [4 \text {$\#$1}^3-\text {$\#$1} a-b\&,3\right ]-\text {Root}\left [4 \text {$\#$1}^3-\text {$\#$1} a-b\&,2\right ]}} \left (y(x)-\text {Root}\left [4 \text {$\#$1}^3-\text {$\#$1} a-b\&,3\right ]\right ) F\left (\text {ArcSin}\left (\sqrt {\frac {\text {Root}\left [4 \text {$\#$1}^3-a \text {$\#$1}-b\&,3\right ]-y(x)}{\text {Root}\left [4 \text {$\#$1}^3-a \text {$\#$1}-b\&,3\right ]-\text {Root}\left [4 \text {$\#$1}^3-a \text {$\#$1}-b\&,2\right ]}}\right )|\frac {\text {Root}\left [4 \text {$\#$1}^3-a \text {$\#$1}-b\&,2\right ]-\text {Root}\left [4 \text {$\#$1}^3-a \text {$\#$1}-b\&,3\right ]}{\text {Root}\left [4 \text {$\#$1}^3-a \text {$\#$1}-b\&,1\right ]-\text {Root}\left [4 \text {$\#$1}^3-a \text {$\#$1}-b\&,3\right ]}\right )}{c_1 \sqrt {2 a y(x)+2 b-8 y(x)^3} \sqrt {\frac {y(x)-\text {Root}\left [4 \text {$\#$1}^3-\text {$\#$1} a-b\&,3\right ]}{\text {Root}\left [4 \text {$\#$1}^3-\text {$\#$1} a-b\&,2\right ]-\text {Root}\left [4 \text {$\#$1}^3-\text {$\#$1} a-b\&,3\right ]}}}=x+c_2,y(x)\right ] \]