Internal problem ID [2342]
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: 5.
ODE order: 1.
ODE degree: 3.
CAS Maple gives this as type [_dAlembert]
\[ \boxed {x {y^{\prime }}^{3}-y y^{\prime }=1} \]
✓ Solution by Maple
Time used: 0.063 (sec). Leaf size: 2049
dsolve(x*diff(y(x),x)^3=y(x)*diff(y(x),x)+1,y(x), singsol=all)
\begin{align*} \frac {12 \left (2 {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}} y \left (x \right )+x \left (\frac {2^{\frac {1}{3}} \left (3^{\frac {1}{6}} \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+3 \,3^{\frac {2}{3}}\right ) {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {1}{3}}}{2}+2^{\frac {2}{3}} 3^{\frac {1}{3}} y \left (x \right )^{2}\right )\right ) x^{3} c_{1} {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}}{\left (y \left (x \right ) 2^{\frac {2}{3}} 3^{\frac {1}{3}} x +{\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}\right )^{2} \left (2^{\frac {2}{3}} 3^{\frac {1}{3}} {\left ({\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right )}^{2} x^{4}\right )}^{\frac {1}{3}}+2 x \left (y \left (x \right ) 3^{\frac {2}{3}} 2^{\frac {1}{3}}-3 {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {1}{3}}\right )\right )^{2}}+x -\frac {18 x^{4} \left (2^{\frac {2}{3}} 3^{\frac {5}{6}} \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}\, x +2 y \left (x \right ) 3^{\frac {2}{3}} 2^{\frac {1}{3}} {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {1}{3}}+9 \,3^{\frac {1}{3}} 2^{\frac {2}{3}} x -3 {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}\right ) 2^{\frac {2}{3}} 3^{\frac {1}{3}} {\left ({\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right )}^{2} x^{4}\right )}^{\frac {1}{3}}}{\left (2 y \left (x \right ) 3^{\frac {2}{3}} 2^{\frac {1}{3}} x +2^{\frac {2}{3}} 3^{\frac {1}{3}} {\left ({\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right )}^{2} x^{4}\right )}^{\frac {1}{3}}-6 x {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {1}{3}}\right )^{2} \left (y \left (x \right ) 2^{\frac {2}{3}} 3^{\frac {1}{3}} x +{\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}\right )^{2}} &= 0 \\ -\frac {3 x^{3} {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}} c_{1} \left (\frac {8 {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}} y \left (x \right )}{9}+x \left (2^{\frac {1}{3}} \left (\left (\frac {i 3^{\frac {2}{3}}}{9}-\frac {3^{\frac {1}{6}}}{9}\right ) \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+i 3^{\frac {1}{6}}-\frac {3^{\frac {2}{3}}}{3}\right ) {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {1}{3}}-\frac {2 y \left (x \right )^{2} 2^{\frac {2}{3}} \left (i 3^{\frac {5}{6}}+3^{\frac {1}{3}}\right )}{9}\right )\right )}{2 {\left (\left (i-\sqrt {3}\right ) {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}+2^{\frac {2}{3}} y \left (x \right ) x \left (i 3^{\frac {1}{3}}+3^{\frac {5}{6}}\right )\right )}^{2} \left (-\frac {2^{\frac {2}{3}} \left (i 3^{\frac {5}{6}}+3^{\frac {1}{3}}\right ) {\left ({\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right )}^{2} x^{4}\right )}^{\frac {1}{3}}}{6}+x \left (-2 {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {1}{3}}+\left (i 3^{\frac {1}{6}}-\frac {3^{\frac {2}{3}}}{3}\right ) y \left (x \right ) 2^{\frac {1}{3}}\right )\right )^{2}}+x +\frac {216 x^{4} 2^{\frac {2}{3}} {\left ({\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right )}^{2} x^{4}\right )}^{\frac {1}{3}} 3^{\frac {1}{3}} \left (-{\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}+y \left (x \right ) \left (i 3^{\frac {1}{6}}-\frac {3^{\frac {2}{3}}}{3}\right ) 2^{\frac {1}{3}} {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {1}{3}}+\frac {x \left (-i 3^{\frac {1}{3}}-\frac {3^{\frac {5}{6}}}{3}\right ) 2^{\frac {2}{3}} \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}}{2}+\frac {3 x \left (-i 3^{\frac {5}{6}}-3^{\frac {1}{3}}\right ) 2^{\frac {2}{3}}}{2}\right )}{{\left (\left (1+i \sqrt {3}\right ) {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}+\left (-i 3^{\frac {5}{6}}+3^{\frac {1}{3}}\right ) x 2^{\frac {2}{3}} y \left (x \right )\right )}^{2} {\left (\frac {\left (-3^{\frac {5}{6}}+i 3^{\frac {1}{3}}\right ) 2^{\frac {2}{3}} {\left ({\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right )}^{2} x^{4}\right )}^{\frac {1}{3}}}{2}+x \left (6 i {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {1}{3}}+2^{\frac {1}{3}} \left (i 3^{\frac {2}{3}}+3 \,3^{\frac {1}{6}}\right ) y \left (x \right )\right )\right )}^{2}} &= 0 \\ \text {Expression too large to display} \\ \end{align*}
✓ Solution by Mathematica
Time used: 144.072 (sec). Leaf size: 21579
DSolve[x*y'[x]^3==y[x]*y'[x]+1,y[x],x,IncludeSingularSolutions -> True]
Too large to display