9.4 problem 223

Internal problem ID [15111]

Book: A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section: Section 8.3. The Lagrange and Clairaut equations. Exercises page 72
Problem number: 223.
ODE order: 1.
ODE degree: 3.

CAS Maple gives this as type [_dAlembert]

\[ \boxed {y-x {y^{\prime }}^{2}+\frac {1}{y^{\prime }}=0} \]

✓ Solution by Maple

Time used: 0.032 (sec). Leaf size: 1835

dsolve(y(x)=x*diff(y(x),x)^2-1/diff(y(x),x),y(x), singsol=all)
 

\begin{align*} \frac {12 x^{3} \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 ) 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 (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 (2^{\frac {1}{3}} 3^{\frac {2}{3}} y \left (x \right )-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 (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 (\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 (\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 ) 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 {2 \left (i 3^{\frac {5}{6}}+3^{\frac {1}{3}}\right ) 2^{\frac {2}{3}} y \left (x \right )^{2}}{9}\right )\right ) {\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}}{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}}+\left (i 3^{\frac {1}{3}}+3^{\frac {5}{6}}\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}+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} {\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}} 2^{\frac {2}{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 y \left (x \right ) 2^{\frac {2}{3}}\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}}+\left (i 3^{\frac {2}{3}}+3 \,3^{\frac {1}{6}}\right ) y \left (x \right ) 2^{\frac {1}{3}}\right )\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 (\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 ) 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 {2 \left (i 3^{\frac {5}{6}}-3^{\frac {1}{3}}\right ) 2^{\frac {2}{3}} y \left (x \right )^{2}}{9}\right )\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}}+2^{\frac {1}{3}} y \left (x \right ) \left (i 3^{\frac {1}{6}}+\frac {3^{\frac {2}{3}}}{3}\right )\right ) x \right )}^{2} {\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}}+2^{\frac {2}{3}} \left (i 3^{\frac {5}{6}}+3^{\frac {1}{3}}\right ) y \left (x \right ) x \right )}^{2}}+x +\frac {216 x^{4} {\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}} 2^{\frac {2}{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 (\frac {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}} \left (i 3^{\frac {1}{3}}+3^{\frac {5}{6}}\right )}{2}+\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}}+y \left (x \right ) \left (i 3^{\frac {2}{3}}-3 \,3^{\frac {1}{6}}\right ) 2^{\frac {1}{3}}\right ) x \right )^{2} {\left ({\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}} \left (\sqrt {3}+i\right )+\left (-3^{\frac {5}{6}}+i 3^{\frac {1}{3}}\right ) x y \left (x \right ) 2^{\frac {2}{3}}\right )}^{2}} &= 0 \\ \end{align*}

✓ Solution by Mathematica

Time used: 145.256 (sec). Leaf size: 19969

DSolve[y[x]==x*y'[x]^2-1/y'[x],y[x],x,IncludeSingularSolutions -> True]
 

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