Internal problem ID [7089]
Book: Own collection of miscellaneous problems
Section: section 1.0
Problem number: 45.
ODE order: 1.
ODE degree: 3.
CAS Maple gives this as type [_dAlembert]
\[ \boxed {y^{\prime } y+x {y^{\prime }}^{3}=1} \]
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 1817
dsolve(diff(y(x),x)*y(x)=1-x*(diff(y(x),x))^3,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 )+\left (\frac {3^{\frac {2}{3}} 2^{\frac {1}{3}} \left (\sqrt {3}\, \sqrt {\frac {4 y \left (x \right )^{3}+27 x}{x}}+9\right ) {\left (\left (\sqrt {3}\, \sqrt {\frac {4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {1}{3}}}{6}+2^{\frac {2}{3}} 3^{\frac {1}{3}} y \left (x \right )^{2}\right ) x \right ) 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}} x^{3}}{\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} \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}}+x -\frac {18 x^{4} \left (\sqrt {\frac {4 y \left (x \right )^{3}+27 x}{x}}\, 2^{\frac {2}{3}} 3^{\frac {5}{6}} x -2 {\left (\left (\sqrt {3}\, \sqrt {\frac {4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {1}{3}} 3^{\frac {2}{3}} 2^{\frac {1}{3}} y \left (x \right )+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 {{\left (\left (\sqrt {3}\, \sqrt {\frac {4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}} \left (-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 )+x \left (\left (\sqrt {3}\, \sqrt {\frac {4 y \left (x \right )^{3}+27 x}{x}}+9\right ) 2^{\frac {1}{3}} \left (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}}-2 y \left (x \right )^{2} \left (i 3^{\frac {5}{6}}+3^{\frac {1}{3}}\right ) 2^{\frac {2}{3}}\right )\right ) c_{1} x^{3}}{6 {\left (\frac {\left (i 3^{\frac {5}{6}}+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}}}{6}+\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}}+y \left (x \right ) 2^{\frac {1}{3}} \left (i 3^{\frac {1}{6}}-\frac {3^{\frac {2}{3}}}{3}\right )\right ) x \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}}+y \left (x \right ) \left (i 3^{\frac {1}{3}}+3^{\frac {5}{6}}\right ) x 2^{\frac {2}{3}}\right )}^{2}}+x +\frac {24 \,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}} x^{4} 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 (-i \sqrt {3}-1\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 ) 2^{\frac {2}{3}} y \left (x \right ) x \right )}^{2} {\left (\frac {\left (i 3^{\frac {5}{6}}+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}}}{6}+\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}}+y \left (x \right ) 2^{\frac {1}{3}} \left (i 3^{\frac {1}{6}}-\frac {3^{\frac {2}{3}}}{3}\right )\right ) x \right )}^{2}} &= 0 \\ \frac {{\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} x^{3} \left (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 )+\left (\left (\sqrt {3}\, \sqrt {\frac {4 y \left (x \right )^{3}+27 x}{x}}+9\right ) 2^{\frac {1}{3}} \left (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}}-2 y \left (x \right )^{2} \left (-3^{\frac {1}{3}}+i 3^{\frac {5}{6}}\right ) 2^{\frac {2}{3}}\right ) x \right )}{6 {\left (\left (i \sqrt {3}-1\right ) {\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 {5}{6}}+3^{\frac {1}{3}}\right ) x 2^{\frac {2}{3}}\right )}^{2} {\left (\frac {\left (-3^{\frac {1}{3}}+i 3^{\frac {5}{6}}\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}}}{6}+\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}}+y \left (x \right ) 2^{\frac {1}{3}} \left (i 3^{\frac {1}{6}}+\frac {3^{\frac {2}{3}}}{3}\right )\right ) x \right )}^{2}}+x +\frac {24 \,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}} x^{4} 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 (-3^{\frac {1}{3}}+i 3^{\frac {5}{6}}\right ) 2^{\frac {2}{3}}}{2}\right )}{{\left (\left (\sqrt {3}+i\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 (3^{\frac {5}{6}}-i 3^{\frac {1}{3}}\right ) 2^{\frac {2}{3}} y \left (x \right ) x \right )}^{2} {\left (\frac {\left (-3^{\frac {1}{3}}+i 3^{\frac {5}{6}}\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}}}{6}+\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}}+y \left (x \right ) 2^{\frac {1}{3}} \left (i 3^{\frac {1}{6}}+\frac {3^{\frac {2}{3}}}{3}\right )\right ) x \right )}^{2}} &= 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 89.497 (sec). Leaf size: 20717
DSolve[y'[x]*y[x]==1-x*(y'[x])^3,y[x],x,IncludeSingularSolutions -> True]
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