75.9.4 problem 223

Internal problem ID [16841]
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
Date solved : Tuesday, January 28, 2025 at 09:34:35 AM
CAS classification : [_dAlembert]

\begin{align*} y&={y^{\prime }}^{2} x -\frac {1}{y^{\prime }} \end{align*}

✓ Solution by Maple

Time used: 0.102 (sec). Leaf size: 2021

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

\begin{align*} \frac {12 \left (2 {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{{2}/{3}} y+x \left (\frac {2^{{1}/{3}} \left (\sqrt {\frac {-4 y^{3}+27 x}{x}}\, 3^{{1}/{6}}+3 \,3^{{2}/{3}}\right ) {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{{1}/{3}}}{2}+2^{{2}/{3}} 3^{{1}/{3}} y^{2}\right )\right ) x^{3} {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{{2}/{3}} c_{1}}{\left (2^{{2}/{3}} 3^{{1}/{3}} {\left ({\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right )}^{2} x^{4}\right )}^{{1}/{3}}+2 x \left (y 3^{{2}/{3}} 2^{{1}/{3}}-3 {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{{1}/{3}}\right )\right )^{2} \left (y 2^{{2}/{3}} 3^{{1}/{3}} x +{\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{{2}/{3}}\right )^{2}}+x -\frac {18 x^{4} \left (\sqrt {\frac {-4 y^{3}+27 x}{x}}\, 3^{{5}/{6}} 2^{{2}/{3}} x +2 y {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{{1}/{3}} 2^{{1}/{3}} 3^{{2}/{3}}+9 \,2^{{2}/{3}} 3^{{1}/{3}} x -3 {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{{2}/{3}}\right ) 2^{{2}/{3}} 3^{{1}/{3}} {\left ({\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right )}^{2} x^{4}\right )}^{{1}/{3}}}{\left (2 y 3^{{2}/{3}} 2^{{1}/{3}} x +2^{{2}/{3}} 3^{{1}/{3}} {\left ({\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right )}^{2} x^{4}\right )}^{{1}/{3}}-6 x {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{{1}/{3}}\right )^{2} \left (y 2^{{2}/{3}} 3^{{1}/{3}} x +{\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{{2}/{3}}\right )^{2}} &= 0 \\ -\frac {3 x^{3} c_{1} {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{{2}/{3}} \left (\frac {8 {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{{2}/{3}} y}{9}+x \left (\left (\left (\frac {i 3^{{2}/{3}}}{9}-\frac {3^{{1}/{6}}}{9}\right ) \sqrt {\frac {-4 y^{3}+27 x}{x}}+i 3^{{1}/{6}}-\frac {3^{{2}/{3}}}{3}\right ) 2^{{1}/{3}} {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{{1}/{3}}-\frac {2 \left (3^{{1}/{3}}+i 3^{{5}/{6}}\right ) 2^{{2}/{3}} y^{2}}{9}\right )\right )}{2 {\left (-\frac {\left (3^{{1}/{3}}+i 3^{{5}/{6}}\right ) 2^{{2}/{3}} {\left ({\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right )}^{2} x^{4}\right )}^{{1}/{3}}}{6}+x \left (-2 {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{{1}/{3}}+2^{{1}/{3}} y \left (i 3^{{1}/{6}}-\frac {3^{{2}/{3}}}{3}\right )\right )\right )}^{2} {\left (\left (i-\sqrt {3}\right ) {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{{2}/{3}}+y \left (3^{{5}/{6}}+i 3^{{1}/{3}}\right ) x 2^{{2}/{3}}\right )}^{2}}+x +\frac {216 x^{4} 2^{{2}/{3}} 3^{{1}/{3}} {\left ({\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right )}^{2} x^{4}\right )}^{{1}/{3}} \left (-{\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{{2}/{3}}+y \left (i 3^{{1}/{6}}-\frac {3^{{2}/{3}}}{3}\right ) 2^{{1}/{3}} {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{{1}/{3}}+\frac {x \left (-i 3^{{1}/{3}}-\frac {3^{{5}/{6}}}{3}\right ) 2^{{2}/{3}} \sqrt {\frac {-4 y^{3}+27 x}{x}}}{2}+\frac {3 x \left (-i 3^{{5}/{6}}-3^{{1}/{3}}\right ) 2^{{2}/{3}}}{2}\right )}{{\left (\left (1+i \sqrt {3}\right ) {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{{2}/{3}}+\left (-i 3^{{5}/{6}}+3^{{1}/{3}}\right ) x 2^{{2}/{3}} y\right )}^{2} {\left (\frac {\left (-3^{{5}/{6}}+i 3^{{1}/{3}}\right ) 2^{{2}/{3}} {\left ({\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right )}^{2} x^{4}\right )}^{{1}/{3}}}{2}+x \left (6 i {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{{1}/{3}}+\left (i 3^{{2}/{3}}+3 \,3^{{1}/{6}}\right ) y 2^{{1}/{3}}\right )\right )}^{2}} &= 0 \\ \frac {c_{1} x^{2} {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{{2}/{3}} \left (2 i \left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2} \sqrt {3}\, 18^{{1}/{3}} {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{{1}/{3}}-12 i \sqrt {3}\, 12^{{1}/{3}} x^{2} y^{2}+2 \left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2} 18^{{1}/{3}} {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{{1}/{3}}+12 \,12^{{1}/{3}} x^{2} y^{2}-48 {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{{2}/{3}} x y\right )}{\left (2 i x y \sqrt {3}\, 18^{{1}/{3}}-i \sqrt {3}\, 12^{{1}/{3}} {\left ({\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right )}^{2} x^{4}\right )}^{{1}/{3}}+y 12^{{2}/{3}} x +12^{{1}/{3}} {\left ({\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right )}^{2} x^{4}\right )}^{{1}/{3}}+12 x {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{{1}/{3}}\right )^{2} \left (i \sqrt {3}\, 12^{{1}/{3}} x y-i \sqrt {3}\, {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{{2}/{3}}+y 12^{{1}/{3}} x +{\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{{2}/{3}}\right )^{2}}+x +\frac {24 x^{2} \left (12 i x^{2} y \sqrt {3}\, 18^{{1}/{3}} {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{{1}/{3}}-54 i \sqrt {3}\, 12^{{1}/{3}} x^{3}+6 \sqrt {3}\, 12^{{1}/{3}} \sqrt {\frac {-4 y^{3}+27 x}{x}}\, x^{3}+12 x^{2} y 18^{{1}/{3}} {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{{1}/{3}}-18 i 12^{{1}/{3}} \sqrt {\frac {-4 y^{3}+27 x}{x}}\, x^{3}+54 \,12^{{1}/{3}} x^{3}+36 x^{2} {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{{2}/{3}}\right ) 12^{{1}/{3}} {\left ({\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right )}^{2} x^{4}\right )}^{{1}/{3}}}{\left (2 i x y \sqrt {3}\, 18^{{1}/{3}}-i \sqrt {3}\, 12^{{1}/{3}} {\left ({\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right )}^{2} x^{4}\right )}^{{1}/{3}}+y 12^{{2}/{3}} x +12^{{1}/{3}} {\left ({\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right )}^{2} x^{4}\right )}^{{1}/{3}}+12 x {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{{1}/{3}}\right )^{2} \left (i \sqrt {3}\, 12^{{1}/{3}} x y-i \sqrt {3}\, {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{{2}/{3}}+y 12^{{1}/{3}} x +{\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{{2}/{3}}\right )^{2}} &= 0 \\ \end{align*}

✓ Solution by Mathematica

Time used: 141.713 (sec). Leaf size: 21579

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

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