2.8 problem 15

Internal problem ID [6040]

Book: Elementary differential equations. By Earl D. Rainville, Phillip E. Bedient. Macmilliam Publishing Co. NY. 6th edition. 1981.
Section: CHAPTER 16. Nonlinear equations. Section 97. The p-discriminant equation. EXERCISES Page 314
Problem number: 15.
ODE order: 1.
ODE degree: 3.

CAS Maple gives this as type [_dAlembert]

Solve \begin {gather*} \boxed {\left (y^{\prime }\right )^{3}+x \left (y^{\prime }\right )^{2}-y=0} \end {gather*}

✓ Solution by Maple

Time used: 0.25 (sec). Leaf size: 1473

dsolve(diff(y(x),x)^3+x*diff(y(x),x)^2-y(x)=0,y(x), singsol=all)
 

\begin{align*} y \relax (x ) = 0 \\ y \relax (x ) = \left (\frac {\left (-36 x^{2}-54 x +108 c_{1}-8 x^{3}+27+6 \sqrt {-48 x^{3} c_{1}-24 x^{3}-216 c_{1} x^{2}-108 x^{2}-324 c_{1} x +324 c_{1}^{2}-162 x +162 c_{1}}\right )^{\frac {1}{3}}}{6}-\frac {6 \left (-\frac {1}{3} x -\frac {1}{9} x^{2}-\frac {1}{4}\right )}{\left (-36 x^{2}-54 x +108 c_{1}-8 x^{3}+27+6 \sqrt {-48 x^{3} c_{1}-24 x^{3}-216 c_{1} x^{2}-108 x^{2}-324 c_{1} x +324 c_{1}^{2}-162 x +162 c_{1}}\right )^{\frac {1}{3}}}-\frac {x}{3}+\frac {1}{2}\right )^{3}+\left (\frac {\left (-36 x^{2}-54 x +108 c_{1}-8 x^{3}+27+6 \sqrt {-48 x^{3} c_{1}-24 x^{3}-216 c_{1} x^{2}-108 x^{2}-324 c_{1} x +324 c_{1}^{2}-162 x +162 c_{1}}\right )^{\frac {1}{3}}}{6}-\frac {6 \left (-\frac {1}{3} x -\frac {1}{9} x^{2}-\frac {1}{4}\right )}{\left (-36 x^{2}-54 x +108 c_{1}-8 x^{3}+27+6 \sqrt {-48 x^{3} c_{1}-24 x^{3}-216 c_{1} x^{2}-108 x^{2}-324 c_{1} x +324 c_{1}^{2}-162 x +162 c_{1}}\right )^{\frac {1}{3}}}-\frac {x}{3}+\frac {1}{2}\right )^{2} x \\ y \relax (x ) = \left (-\frac {\left (-36 x^{2}-54 x +108 c_{1}-8 x^{3}+27+6 \sqrt {-48 x^{3} c_{1}-24 x^{3}-216 c_{1} x^{2}-108 x^{2}-324 c_{1} x +324 c_{1}^{2}-162 x +162 c_{1}}\right )^{\frac {1}{3}}}{12}+\frac {-x -\frac {1}{3} x^{2}-\frac {3}{4}}{\left (-36 x^{2}-54 x +108 c_{1}-8 x^{3}+27+6 \sqrt {-48 x^{3} c_{1}-24 x^{3}-216 c_{1} x^{2}-108 x^{2}-324 c_{1} x +324 c_{1}^{2}-162 x +162 c_{1}}\right )^{\frac {1}{3}}}-\frac {x}{3}+\frac {1}{2}-\frac {i \sqrt {3}\, \left (\frac {\left (-36 x^{2}-54 x +108 c_{1}-8 x^{3}+27+6 \sqrt {-48 x^{3} c_{1}-24 x^{3}-216 c_{1} x^{2}-108 x^{2}-324 c_{1} x +324 c_{1}^{2}-162 x +162 c_{1}}\right )^{\frac {1}{3}}}{6}+\frac {-2 x -\frac {2}{3} x^{2}-\frac {3}{2}}{\left (-36 x^{2}-54 x +108 c_{1}-8 x^{3}+27+6 \sqrt {-48 x^{3} c_{1}-24 x^{3}-216 c_{1} x^{2}-108 x^{2}-324 c_{1} x +324 c_{1}^{2}-162 x +162 c_{1}}\right )^{\frac {1}{3}}}\right )}{2}\right )^{3}+\left (-\frac {\left (-36 x^{2}-54 x +108 c_{1}-8 x^{3}+27+6 \sqrt {-48 x^{3} c_{1}-24 x^{3}-216 c_{1} x^{2}-108 x^{2}-324 c_{1} x +324 c_{1}^{2}-162 x +162 c_{1}}\right )^{\frac {1}{3}}}{12}+\frac {-x -\frac {1}{3} x^{2}-\frac {3}{4}}{\left (-36 x^{2}-54 x +108 c_{1}-8 x^{3}+27+6 \sqrt {-48 x^{3} c_{1}-24 x^{3}-216 c_{1} x^{2}-108 x^{2}-324 c_{1} x +324 c_{1}^{2}-162 x +162 c_{1}}\right )^{\frac {1}{3}}}-\frac {x}{3}+\frac {1}{2}-\frac {i \sqrt {3}\, \left (\frac {\left (-36 x^{2}-54 x +108 c_{1}-8 x^{3}+27+6 \sqrt {-48 x^{3} c_{1}-24 x^{3}-216 c_{1} x^{2}-108 x^{2}-324 c_{1} x +324 c_{1}^{2}-162 x +162 c_{1}}\right )^{\frac {1}{3}}}{6}+\frac {-2 x -\frac {2}{3} x^{2}-\frac {3}{2}}{\left (-36 x^{2}-54 x +108 c_{1}-8 x^{3}+27+6 \sqrt {-48 x^{3} c_{1}-24 x^{3}-216 c_{1} x^{2}-108 x^{2}-324 c_{1} x +324 c_{1}^{2}-162 x +162 c_{1}}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2} x \\ y \relax (x ) = \left (-\frac {\left (-36 x^{2}-54 x +108 c_{1}-8 x^{3}+27+6 \sqrt {-48 x^{3} c_{1}-24 x^{3}-216 c_{1} x^{2}-108 x^{2}-324 c_{1} x +324 c_{1}^{2}-162 x +162 c_{1}}\right )^{\frac {1}{3}}}{12}+\frac {-x -\frac {1}{3} x^{2}-\frac {3}{4}}{\left (-36 x^{2}-54 x +108 c_{1}-8 x^{3}+27+6 \sqrt {-48 x^{3} c_{1}-24 x^{3}-216 c_{1} x^{2}-108 x^{2}-324 c_{1} x +324 c_{1}^{2}-162 x +162 c_{1}}\right )^{\frac {1}{3}}}-\frac {x}{3}+\frac {1}{2}+\frac {i \sqrt {3}\, \left (\frac {\left (-36 x^{2}-54 x +108 c_{1}-8 x^{3}+27+6 \sqrt {-48 x^{3} c_{1}-24 x^{3}-216 c_{1} x^{2}-108 x^{2}-324 c_{1} x +324 c_{1}^{2}-162 x +162 c_{1}}\right )^{\frac {1}{3}}}{6}+\frac {-2 x -\frac {2}{3} x^{2}-\frac {3}{2}}{\left (-36 x^{2}-54 x +108 c_{1}-8 x^{3}+27+6 \sqrt {-48 x^{3} c_{1}-24 x^{3}-216 c_{1} x^{2}-108 x^{2}-324 c_{1} x +324 c_{1}^{2}-162 x +162 c_{1}}\right )^{\frac {1}{3}}}\right )}{2}\right )^{3}+\left (-\frac {\left (-36 x^{2}-54 x +108 c_{1}-8 x^{3}+27+6 \sqrt {-48 x^{3} c_{1}-24 x^{3}-216 c_{1} x^{2}-108 x^{2}-324 c_{1} x +324 c_{1}^{2}-162 x +162 c_{1}}\right )^{\frac {1}{3}}}{12}+\frac {-x -\frac {1}{3} x^{2}-\frac {3}{4}}{\left (-36 x^{2}-54 x +108 c_{1}-8 x^{3}+27+6 \sqrt {-48 x^{3} c_{1}-24 x^{3}-216 c_{1} x^{2}-108 x^{2}-324 c_{1} x +324 c_{1}^{2}-162 x +162 c_{1}}\right )^{\frac {1}{3}}}-\frac {x}{3}+\frac {1}{2}+\frac {i \sqrt {3}\, \left (\frac {\left (-36 x^{2}-54 x +108 c_{1}-8 x^{3}+27+6 \sqrt {-48 x^{3} c_{1}-24 x^{3}-216 c_{1} x^{2}-108 x^{2}-324 c_{1} x +324 c_{1}^{2}-162 x +162 c_{1}}\right )^{\frac {1}{3}}}{6}+\frac {-2 x -\frac {2}{3} x^{2}-\frac {3}{2}}{\left (-36 x^{2}-54 x +108 c_{1}-8 x^{3}+27+6 \sqrt {-48 x^{3} c_{1}-24 x^{3}-216 c_{1} x^{2}-108 x^{2}-324 c_{1} x +324 c_{1}^{2}-162 x +162 c_{1}}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2} x \\ \end{align*}

✓ Solution by Mathematica

Time used: 99.936 (sec). Leaf size: 1410

DSolve[(y'[x])^3+x*(y'[x])^2-y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {1}{24} \left (-4 x^2+2 x \left (6+\sqrt [3]{-2 x (2 x (2 x+9)+27)+3 \left (2 \sqrt {6} \sqrt {-((1+2 c_1) (x (2 x (2 x+9)+27)-27 c_1))}+9+36 c_1\right )}\right )+3 \left (9+\sqrt [3]{-2 x (2 x (2 x+9)+27)+3 \left (2 \sqrt {6} \sqrt {-((1+2 c_1) (x (2 x (2 x+9)+27)-27 c_1))}+9+36 c_1\right )}\right )+\frac {24 c_1 (2 x+3)^3-(2 x+3)^3 \left (-2 x (2 x (2 x+9)+27)+3 \left (2 \sqrt {6} \sqrt {-((1+2 c_1) (x (2 x (2 x+9)+27)-27 c_1))}+9+36 c_1\right )\right ){}^{2/3}+108 c_1 (2 x+3) \sqrt [3]{-2 x (2 x (2 x+9)+27)+3 \left (2 \sqrt {6} \sqrt {-((1+2 c_1) (x (2 x (2 x+9)+27)-27 c_1))}+9+36 c_1\right )}-6 \sqrt {6} (2 x+3) \sqrt {-((1+2 c_1) (x (2 x (2 x+9)+27)-27 c_1))} \sqrt [3]{-2 x (2 x (2 x+9)+27)+3 \left (2 \sqrt {6} \sqrt {-((1+2 c_1) (x (2 x (2 x+9)+27)-27 c_1))}+9+36 c_1\right )}+54 (2 x+3) \sqrt [3]{-2 x (2 x (2 x+9)+27)+3 \left (2 \sqrt {6} \sqrt {-((1+2 c_1) (x (2 x (2 x+9)+27)-27 c_1))}+9+36 c_1\right )}+216 c_1 \left (-2 x (2 x (2 x+9)+27)+3 \left (2 \sqrt {6} \sqrt {-((1+2 c_1) (x (2 x (2 x+9)+27)-27 c_1))}+9+36 c_1\right )\right ){}^{2/3}-12 \sqrt {6} \sqrt {-((1+2 c_1) (x (2 x (2 x+9)+27)-27 c_1))} \left (-2 x (2 x (2 x+9)+27)+3 \left (2 \sqrt {6} \sqrt {-((1+2 c_1) (x (2 x (2 x+9)+27)-27 c_1))}+9+36 c_1\right )\right ){}^{2/3}+108 \left (-2 x (2 x (2 x+9)+27)+3 \left (2 \sqrt {6} \sqrt {-((1+2 c_1) (x (2 x (2 x+9)+27)-27 c_1))}+9+36 c_1\right )\right ){}^{2/3}}{(2 x+3)^3}\right ) \\ y(x)\to \frac {1}{6} \left (2 (3-2 x) x-6 x-\frac {i \left (\sqrt {3}-i\right ) x (2 x+3)^2}{\sqrt [3]{-2 x (2 x (2 x+9)+27)+3 \left (2 \sqrt {6} \sqrt {-((1+2 c_1) (x (2 x (2 x+9)+27)-27 c_1))}+9+36 c_1\right )}}+\frac {1}{16} \left (-4 x-\frac {i \left (\sqrt {3}-i\right ) (2 x+3)^2}{\sqrt [3]{-2 x (2 x (2 x+9)+27)+3 \left (2 \sqrt {6} \sqrt {-((1+2 c_1) (x (2 x (2 x+9)+27)-27 c_1))}+9+36 c_1\right )}}+i \left (\sqrt {3}+i\right ) \sqrt [3]{-2 x (2 x (2 x+9)+27)+3 \left (2 \sqrt {6} \sqrt {-((1+2 c_1) (x (2 x (2 x+9)+27)-27 c_1))}+9+36 c_1\right )}+6\right ){}^2+i \left (\sqrt {3}+i\right ) x \sqrt [3]{-2 x (2 x (2 x+9)+27)+3 \left (2 \sqrt {6} \sqrt {-((1+2 c_1) (x (2 x (2 x+9)+27)-27 c_1))}+9+36 c_1\right )}+6 c_1\right ) \\ y(x)\to \frac {1}{6} \left (2 (3-2 x) x-6 x+\frac {i \left (\sqrt {3}+i\right ) x (2 x+3)^2}{\sqrt [3]{-2 x (2 x (2 x+9)+27)+3 \left (2 \sqrt {6} \sqrt {-((1+2 c_1) (x (2 x (2 x+9)+27)-27 c_1))}+9+36 c_1\right )}}+\frac {1}{16} \left (4 x+\frac {\left (1-i \sqrt {3}\right ) (2 x+3)^2}{\sqrt [3]{-2 x (2 x (2 x+9)+27)+3 \left (2 \sqrt {6} \sqrt {-((1+2 c_1) (x (2 x (2 x+9)+27)-27 c_1))}+9+36 c_1\right )}}+\left (1+i \sqrt {3}\right ) \sqrt [3]{-2 x (2 x (2 x+9)+27)+3 \left (2 \sqrt {6} \sqrt {-((1+2 c_1) (x (2 x (2 x+9)+27)-27 c_1))}+9+36 c_1\right )}-6\right ){}^2-\left (1+i \sqrt {3}\right ) x \sqrt [3]{-2 x (2 x (2 x+9)+27)+3 \left (2 \sqrt {6} \sqrt {-((1+2 c_1) (x (2 x (2 x+9)+27)-27 c_1))}+9+36 c_1\right )}+6 c_1\right ) \\ \end{align*}