15.20.5 problem 5

Internal problem ID [3313]
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
Date solved : Monday, January 27, 2025 at 07:33:15 AM
CAS classification : [_dAlembert]

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

✓ Solution by Maple

Time used: 0.038 (sec). Leaf size: 1807

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

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

✓ Solution by Mathematica

Time used: 142.190 (sec). Leaf size: 21579

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

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