2.8 problem 15

Internal problem ID [6793]

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]

\[ \boxed {{y^{\prime }}^{3}+x {y^{\prime }}^{2}-y=0} \]

✓ Solution by Maple

Time used: 0.078 (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 \left (x \right ) = 0 y \left (x \right ) = {\left (\frac {\left (-36 x^{2}-54 x +108 c_{1} -8 x^{3}+27+6 \sqrt {-48 c_{1} x^{3}-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 c_{1} x^{3}-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}+x {\left (\frac {\left (-36 x^{2}-54 x +108 c_{1} -8 x^{3}+27+6 \sqrt {-48 c_{1} x^{3}-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 c_{1} x^{3}-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} y \left (x \right ) = {\left (-\frac {\left (-36 x^{2}-54 x +108 c_{1} -8 x^{3}+27+6 \sqrt {-48 c_{1} x^{3}-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 c_{1} x^{3}-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 c_{1} x^{3}-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 c_{1} x^{3}-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}+x {\left (-\frac {\left (-36 x^{2}-54 x +108 c_{1} -8 x^{3}+27+6 \sqrt {-48 c_{1} x^{3}-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 c_{1} x^{3}-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 c_{1} x^{3}-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 c_{1} x^{3}-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} y \left (x \right ) = {\left (-\frac {\left (-36 x^{2}-54 x +108 c_{1} -8 x^{3}+27+6 \sqrt {-48 c_{1} x^{3}-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 c_{1} x^{3}-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 c_{1} x^{3}-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 c_{1} x^{3}-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}+x {\left (-\frac {\left (-36 x^{2}-54 x +108 c_{1} -8 x^{3}+27+6 \sqrt {-48 c_{1} x^{3}-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 c_{1} x^{3}-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 c_{1} x^{3}-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 c_{1} x^{3}-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} \end{align*}

✓ Solution by Mathematica

Time used: 84.497 (sec). Leaf size: 1516

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

\begin{align*} y(x)\to \frac {-16 x^4+8 \left (\sqrt [3]{-8 x^3-36 x^2-54 x+108 c_1+6 \sqrt {6} \sqrt {-\left (\left (4 x^3+18 x^2+27 x-27 c_1\right ) (2 c_1+1)\right )}+27}-12\right ) x^3-4 \left (\left (-8 x^3-36 x^2-54 x+108 c_1+6 \sqrt {6} \sqrt {-\left (\left (4 x^3+18 x^2+27 x-27 c_1\right ) (2 c_1+1)\right )}+27\right ){}^{2/3}-9 \sqrt [3]{-8 x^3-36 x^2-54 x+108 c_1+6 \sqrt {6} \sqrt {-\left (\left (4 x^3+18 x^2+27 x-27 c_1\right ) (2 c_1+1)\right )}+27}+54\right ) x^2+6 \left (72 c_1+2 \left (-8 x^3-36 x^2-54 x+108 c_1+6 \sqrt {6} \sqrt {-\left (\left (4 x^3+18 x^2+27 x-27 c_1\right ) (2 c_1+1)\right )}+27\right ){}^{2/3}+4 \sqrt {6} \sqrt {-\left (\left (4 x^3+18 x^2+27 x-27 c_1\right ) (2 c_1+1)\right )}+9 \sqrt [3]{-8 x^3-36 x^2-54 x+108 c_1+6 \sqrt {6} \sqrt {-\left (\left (4 x^3+18 x^2+27 x-27 c_1\right ) (2 c_1+1)\right )}+27}\right ) x+3 \left (4 c_1 \left (2 \left (-8 x^3-36 x^2-54 x+108 c_1+6 \sqrt {6} \sqrt {-\left (\left (4 x^3+18 x^2+27 x-27 c_1\right ) (2 c_1+1)\right )}+27\right ){}^{2/3}+9 \sqrt [3]{-8 x^3-36 x^2-54 x+108 c_1+6 \sqrt {6} \sqrt {-\left (\left (4 x^3+18 x^2+27 x-27 c_1\right ) (2 c_1+1)\right )}+27}+54\right )+9 \left (-8 x^3-36 x^2-54 x+108 c_1+6 \sqrt {6} \sqrt {-\left (\left (4 x^3+18 x^2+27 x-27 c_1\right ) (2 c_1+1)\right )}+27\right ){}^{2/3}+12 \sqrt {6} \sqrt {-\left (\left (4 x^3+18 x^2+27 x-27 c_1\right ) (2 c_1+1)\right )}+2 \sqrt {6} \sqrt {-\left (\left (4 x^3+18 x^2+27 x-27 c_1\right ) (2 c_1+1)\right )} \sqrt [3]{-8 x^3-36 x^2-54 x+108 c_1+6 \sqrt {6} \sqrt {-\left (\left (4 x^3+18 x^2+27 x-27 c_1\right ) (2 c_1+1)\right )}+27}+27 \sqrt [3]{-8 x^3-36 x^2-54 x+108 c_1+6 \sqrt {6} \sqrt {-\left (\left (4 x^3+18 x^2+27 x-27 c_1\right ) (2 c_1+1)\right )}+27}+81\right )}{24 \left (-8 x^3-36 x^2-54 x+108 c_1+6 \sqrt {6} \sqrt {-\left (\left (4 x^3+18 x^2+27 x-27 c_1\right ) (2 c_1+1)\right )}+27\right ){}^{2/3}} y(x)\to \frac {1}{6} \left (-\frac {i \left (\sqrt {3}-i\right ) x (2 x+3)^2}{\sqrt [3]{-8 x^3-36 x^2+6 \sqrt {6} \sqrt {-\left ((1+2 c_1) \left (4 x^3+18 x^2+27 x-27 c_1\right )\right )}-54 x+27+108 c_1}}+\frac {1}{16} \left (-\frac {i \left (\sqrt {3}-i\right ) (2 x+3)^2}{\sqrt [3]{-8 x^3-36 x^2+6 \sqrt {6} \sqrt {-\left ((1+2 c_1) \left (4 x^3+18 x^2+27 x-27 c_1\right )\right )}-54 x+27+108 c_1}}+i \left (\sqrt {3}+i\right ) \sqrt [3]{-8 x^3-36 x^2+6 \sqrt {6} \sqrt {-\left ((1+2 c_1) \left (4 x^3+18 x^2+27 x-27 c_1\right )\right )}-54 x+27+108 c_1}-4 x+6\right ){}^2+i \left (\sqrt {3}+i\right ) x \sqrt [3]{-8 x^3-36 x^2+6 \sqrt {6} \sqrt {-\left ((1+2 c_1) \left (4 x^3+18 x^2+27 x-27 c_1\right )\right )}-54 x+27+108 c_1}+2 (3-2 x) x-6 x+6 c_1\right ) y(x)\to \frac {1}{6} \left (\frac {i \left (\sqrt {3}+i\right ) x (2 x+3)^2}{\sqrt [3]{-8 x^3-36 x^2+6 \sqrt {6} \sqrt {-\left ((1+2 c_1) \left (4 x^3+18 x^2+27 x-27 c_1\right )\right )}-54 x+27+108 c_1}}+\frac {1}{16} \left (\frac {\left (1-i \sqrt {3}\right ) (2 x+3)^2}{\sqrt [3]{-8 x^3-36 x^2+6 \sqrt {6} \sqrt {-\left ((1+2 c_1) \left (4 x^3+18 x^2+27 x-27 c_1\right )\right )}-54 x+27+108 c_1}}+\left (1+i \sqrt {3}\right ) \sqrt [3]{-8 x^3-36 x^2+6 \sqrt {6} \sqrt {-\left ((1+2 c_1) \left (4 x^3+18 x^2+27 x-27 c_1\right )\right )}-54 x+27+108 c_1}+4 x-6\right ){}^2-\left (1+i \sqrt {3}\right ) x \sqrt [3]{-8 x^3-36 x^2+6 \sqrt {6} \sqrt {-\left ((1+2 c_1) \left (4 x^3+18 x^2+27 x-27 c_1\right )\right )}-54 x+27+108 c_1}+2 (3-2 x) x-6 x+6 c_1\right ) \end{align*}