56.1.45 problem 45

Internal problem ID [8757]
Book : Own collection of miscellaneous problems
Section : section 1.0
Problem number : 45
Date solved : Monday, January 27, 2025 at 04:48:20 PM
CAS classification : [_dAlembert]

\begin{align*} y^{\prime } y&=1-x {y^{\prime }}^{3} \end{align*}

✓ Solution by Maple

Time used: 0.035 (sec). Leaf size: 1798

dsolve(diff(y(x),x)*y(x)=1-x*(diff(y(x),x))^3,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 86.107 (sec). Leaf size: 20717

DSolve[D[y[x],x]*y[x]==1-x*(D[y[x],x])^3,y[x],x,IncludeSingularSolutions -> True]
 

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