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: 0.15 (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}-216 c_{1} x^{2}-24 x^{3}+324 c_{1}^{2}-324 c_{1} x -108 x^{2}+162 c_{1}-162 x}\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}-216 c_{1} x^{2}-24 x^{3}+324 c_{1}^{2}-324 c_{1} x -108 x^{2}+162 c_{1}-162 x}\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}-216 c_{1} x^{2}-24 x^{3}+324 c_{1}^{2}-324 c_{1} x -108 x^{2}+162 c_{1}-162 x}\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}-216 c_{1} x^{2}-24 x^{3}+324 c_{1}^{2}-324 c_{1} x -108 x^{2}+162 c_{1}-162 x}\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}-216 c_{1} x^{2}-24 x^{3}+324 c_{1}^{2}-324 c_{1} x -108 x^{2}+162 c_{1}-162 x}\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}-216 c_{1} x^{2}-24 x^{3}+324 c_{1}^{2}-324 c_{1} x -108 x^{2}+162 c_{1}-162 x}\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}-216 c_{1} x^{2}-24 x^{3}+324 c_{1}^{2}-324 c_{1} x -108 x^{2}+162 c_{1}-162 x}\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}-216 c_{1} x^{2}-24 x^{3}+324 c_{1}^{2}-324 c_{1} x -108 x^{2}+162 c_{1}-162 x}\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}-216 c_{1} x^{2}-24 x^{3}+324 c_{1}^{2}-324 c_{1} x -108 x^{2}+162 c_{1}-162 x}\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}-216 c_{1} x^{2}-24 x^{3}+324 c_{1}^{2}-324 c_{1} x -108 x^{2}+162 c_{1}-162 x}\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}-216 c_{1} x^{2}-24 x^{3}+324 c_{1}^{2}-324 c_{1} x -108 x^{2}+162 c_{1}-162 x}\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}-216 c_{1} x^{2}-24 x^{3}+324 c_{1}^{2}-324 c_{1} x -108 x^{2}+162 c_{1}-162 x}\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}-216 c_{1} x^{2}-24 x^{3}+324 c_{1}^{2}-324 c_{1} x -108 x^{2}+162 c_{1}-162 x}\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}-216 c_{1} x^{2}-24 x^{3}+324 c_{1}^{2}-324 c_{1} x -108 x^{2}+162 c_{1}-162 x}\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}-216 c_{1} x^{2}-24 x^{3}+324 c_{1}^{2}-324 c_{1} x -108 x^{2}+162 c_{1}-162 x}\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}-216 c_{1} x^{2}-24 x^{3}+324 c_{1}^{2}-324 c_{1} x -108 x^{2}+162 c_{1}-162 x}\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}-216 c_{1} x^{2}-24 x^{3}+324 c_{1}^{2}-324 c_{1} x -108 x^{2}+162 c_{1}-162 x}\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}-216 c_{1} x^{2}-24 x^{3}+324 c_{1}^{2}-324 c_{1} x -108 x^{2}+162 c_{1}-162 x}\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}-216 c_{1} x^{2}-24 x^{3}+324 c_{1}^{2}-324 c_{1} x -108 x^{2}+162 c_{1}-162 x}\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}-216 c_{1} x^{2}-24 x^{3}+324 c_{1}^{2}-324 c_{1} x -108 x^{2}+162 c_{1}-162 x}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2} x \\ \end{align*}

✓ Solution by Mathematica

Time used: 90.273 (sec). Leaf size: 958

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

\begin{align*} y(x)\to \frac {-144 x^4+72 x^3 \left (-12+\sqrt [3]{-(2 x+3)^3+\sqrt {c_1 \left (-2 (2 x+3)^3+c_1\right )}+c_1}\right )-36 x^2 \left (\left (-(2 x+3)^3+\sqrt {c_1 \left (-2 (2 x+3)^3+c_1\right )}+c_1\right ){}^{2/3}-9 \sqrt [3]{-(2 x+3)^3+\sqrt {c_1 \left (-2 (2 x+3)^3+c_1\right )}+c_1}+54\right )+18 x \left (6 \left (-(2 x+3)^3+\sqrt {c_1 \left (-2 (2 x+3)^3+c_1\right )}+c_1\right ){}^{2/3}+2 \sqrt {c_1 \left (-2 (2 x+3)^3+c_1\right )}+27 \sqrt [3]{-(2 x+3)^3+\sqrt {c_1 \left (-2 (2 x+3)^3+c_1\right )}+c_1}-108+2 c_1\right )+27 \left (2 \sqrt {c_1 \left (-2 (2 x+3)^3+c_1\right )}+9 \sqrt [3]{-(2 x+3)^3+\sqrt {c_1 \left (-2 (2 x+3)^3+c_1\right )}+c_1}-27+2 c_1\right )+\sqrt [3]{-(2 x+3)^3+\sqrt {c_1 \left (-2 (2 x+3)^3+c_1\right )}+c_1} \left (c_1 \left (9+2 \sqrt [3]{-(2 x+3)^3+\sqrt {c_1 \left (-2 (2 x+3)^3+c_1\right )}+c_1}\right )+9 \left (\sqrt {c_1 \left (-2 (2 x+3)^3+c_1\right )}+15 \sqrt [3]{-(2 x+3)^3+\sqrt {c_1 \left (-2 (2 x+3)^3+c_1\right )}+c_1}\right )\right )}{216 \left (-(2 x+3)^3+\sqrt {c_1 \left (-2 (2 x+3)^3+c_1\right )}+c_1\right ){}^{2/3}} \\ y(x)\to \frac {1}{864} \left (-576 x^2+144 i \left (\sqrt {3}+i\right ) x \sqrt [3]{-(2 x+3)^3+\sqrt {c_1 \left (-2 (2 x+3)^3+c_1\right )}+c_1}-\frac {144 i \left (\sqrt {3}-i\right ) (2 x+3)^2 x}{\sqrt [3]{-(2 x+3)^3+\sqrt {c_1 \left (-2 (2 x+3)^3+c_1\right )}+c_1}}+9 \left (-4 x-\frac {i \left (\sqrt {3}-i\right ) (2 x+3)^2}{\sqrt [3]{-(2 x+3)^3+\sqrt {c_1 \left (-2 (2 x+3)^3+c_1\right )}+c_1}}+i \left (\sqrt {3}+i\right ) \sqrt [3]{-(2 x+3)^3+\sqrt {c_1 \left (-2 (2 x+3)^3+c_1\right )}+c_1}+6\right ){}^2+8 (-54+c_1)\right ) \\ y(x)\to \frac {1}{864} \left (-576 x^2-144 i \left (\sqrt {3}-i\right ) x \sqrt [3]{-(2 x+3)^3+\sqrt {c_1 \left (-2 (2 x+3)^3+c_1\right )}+c_1}+\frac {144 i \left (\sqrt {3}+i\right ) (2 x+3)^2 x}{\sqrt [3]{-(2 x+3)^3+\sqrt {c_1 \left (-2 (2 x+3)^3+c_1\right )}+c_1}}+9 \left (-4 x+\frac {i \left (\sqrt {3}+i\right ) (2 x+3)^2}{\sqrt [3]{-(2 x+3)^3+\sqrt {c_1 \left (-2 (2 x+3)^3+c_1\right )}+c_1}}-i \left (\sqrt {3}-i\right ) \sqrt [3]{-(2 x+3)^3+\sqrt {c_1 \left (-2 (2 x+3)^3+c_1\right )}+c_1}+6\right ){}^2+8 (-54+c_1)\right ) \\ y(x)\to 0 \\ \end{align*}