Internal problem ID [2348]
Book: Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath.
Boston. 1964
Section: Exercise 38, page 173
Problem number: 11.
ODE order: 1.
ODE degree: 3.
CAS Maple gives this as type [[_1st_order, _with_linear_symmetries], _dAlembert]
\[ \boxed {2 x {y^{\prime }}^{3}-y {y^{\prime }}^{2}=-1} \]
✓ Solution by Maple
Time used: 0.234 (sec). Leaf size: 620
dsolve(2*diff(y(x),x)^3*x+1=diff(y(x),x)^2*y(x),y(x), singsol=all)
\begin{align*} y \left (x \right ) = 2 x \left (\frac {{\left (\left (-27+3 \sqrt {-\frac {3 \left (c_{1}^{3}-27 x \right )}{x}}\right ) x^{2}\right )}^{\frac {1}{3}}}{3 x}+\frac {c_{1}}{{\left (\left (-27+3 \sqrt {-\frac {3 \left (c_{1}^{3}-27 x \right )}{x}}\right ) x^{2}\right )}^{\frac {1}{3}}}\right )+\frac {1}{{\left (\frac {{\left (\left (-27+3 \sqrt {-\frac {3 \left (c_{1}^{3}-27 x \right )}{x}}\right ) x^{2}\right )}^{\frac {1}{3}}}{3 x}+\frac {c_{1}}{{\left (\left (-27+3 \sqrt {-\frac {3 \left (c_{1}^{3}-27 x \right )}{x}}\right ) x^{2}\right )}^{\frac {1}{3}}}\right )}^{2}} y \left (x \right ) = 2 x \left (-\frac {{\left (\left (-27+3 \sqrt {-\frac {3 \left (c_{1}^{3}-27 x \right )}{x}}\right ) x^{2}\right )}^{\frac {1}{3}}}{6 x}-\frac {c_{1}}{2 {\left (\left (-27+3 \sqrt {-\frac {3 \left (c_{1}^{3}-27 x \right )}{x}}\right ) x^{2}\right )}^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {{\left (\left (-27+3 \sqrt {-\frac {3 \left (c_{1}^{3}-27 x \right )}{x}}\right ) x^{2}\right )}^{\frac {1}{3}}}{3 x}-\frac {c_{1}}{{\left (\left (-27+3 \sqrt {-\frac {3 \left (c_{1}^{3}-27 x \right )}{x}}\right ) x^{2}\right )}^{\frac {1}{3}}}\right )}{2}\right )+\frac {1}{{\left (-\frac {{\left (\left (-27+3 \sqrt {-\frac {3 \left (c_{1}^{3}-27 x \right )}{x}}\right ) x^{2}\right )}^{\frac {1}{3}}}{6 x}-\frac {c_{1}}{2 {\left (\left (-27+3 \sqrt {-\frac {3 \left (c_{1}^{3}-27 x \right )}{x}}\right ) x^{2}\right )}^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {{\left (\left (-27+3 \sqrt {-\frac {3 \left (c_{1}^{3}-27 x \right )}{x}}\right ) x^{2}\right )}^{\frac {1}{3}}}{3 x}-\frac {c_{1}}{{\left (\left (-27+3 \sqrt {-\frac {3 \left (c_{1}^{3}-27 x \right )}{x}}\right ) x^{2}\right )}^{\frac {1}{3}}}\right )}{2}\right )}^{2}} y \left (x \right ) = 2 x \left (-\frac {{\left (\left (-27+3 \sqrt {-\frac {3 \left (c_{1}^{3}-27 x \right )}{x}}\right ) x^{2}\right )}^{\frac {1}{3}}}{6 x}-\frac {c_{1}}{2 {\left (\left (-27+3 \sqrt {-\frac {3 \left (c_{1}^{3}-27 x \right )}{x}}\right ) x^{2}\right )}^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {{\left (\left (-27+3 \sqrt {-\frac {3 \left (c_{1}^{3}-27 x \right )}{x}}\right ) x^{2}\right )}^{\frac {1}{3}}}{3 x}-\frac {c_{1}}{{\left (\left (-27+3 \sqrt {-\frac {3 \left (c_{1}^{3}-27 x \right )}{x}}\right ) x^{2}\right )}^{\frac {1}{3}}}\right )}{2}\right )+\frac {1}{{\left (-\frac {{\left (\left (-27+3 \sqrt {-\frac {3 \left (c_{1}^{3}-27 x \right )}{x}}\right ) x^{2}\right )}^{\frac {1}{3}}}{6 x}-\frac {c_{1}}{2 {\left (\left (-27+3 \sqrt {-\frac {3 \left (c_{1}^{3}-27 x \right )}{x}}\right ) x^{2}\right )}^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {{\left (\left (-27+3 \sqrt {-\frac {3 \left (c_{1}^{3}-27 x \right )}{x}}\right ) x^{2}\right )}^{\frac {1}{3}}}{3 x}-\frac {c_{1}}{{\left (\left (-27+3 \sqrt {-\frac {3 \left (c_{1}^{3}-27 x \right )}{x}}\right ) x^{2}\right )}^{\frac {1}{3}}}\right )}{2}\right )}^{2}} \end{align*}
✓ Solution by Mathematica
Time used: 151.15 (sec). Leaf size: 17695
DSolve[2*y'[x]^3*x+1==y'[x]^2*y[x],y[x],x,IncludeSingularSolutions -> True]
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