Internal problem ID [8864]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 529.
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: 0.062 (sec). Leaf size: 994
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 ) &= \frac {\left (4 x^{2}-2 x \left (-36 x^{2}-54 x +108 c_{1} -8 x^{3}+27+6 \sqrt {-6 \left (1+2 c_{1} \right ) \left (4 x^{3}+18 x^{2}-27 c_{1} +27 x \right )}\right )^{\frac {1}{3}}+12 x +3 \left (-36 x^{2}-54 x +108 c_{1} -8 x^{3}+27+6 \sqrt {-6 \left (1+2 c_{1} \right ) \left (4 x^{3}+18 x^{2}-27 c_{1} +27 x \right )}\right )^{\frac {1}{3}}+\left (-36 x^{2}-54 x +108 c_{1} -8 x^{3}+27+6 \sqrt {-6 \left (1+2 c_{1} \right ) \left (4 x^{3}+18 x^{2}-27 c_{1} +27 x \right )}\right )^{\frac {2}{3}}+9\right )^{2} \left (4 x^{2}+4 x \left (-36 x^{2}-54 x +108 c_{1} -8 x^{3}+27+6 \sqrt {-6 \left (1+2 c_{1} \right ) \left (4 x^{3}+18 x^{2}-27 c_{1} +27 x \right )}\right )^{\frac {1}{3}}+12 x +3 \left (-36 x^{2}-54 x +108 c_{1} -8 x^{3}+27+6 \sqrt {-6 \left (1+2 c_{1} \right ) \left (4 x^{3}+18 x^{2}-27 c_{1} +27 x \right )}\right )^{\frac {1}{3}}+\left (-36 x^{2}-54 x +108 c_{1} -8 x^{3}+27+6 \sqrt {-6 \left (1+2 c_{1} \right ) \left (4 x^{3}+18 x^{2}-27 c_{1} +27 x \right )}\right )^{\frac {2}{3}}+9\right )}{-1728 x^{3}-7776 x^{2}-11664 x +23328 c_{1} +1296 \sqrt {-6 \left (1+2 c_{1} \right ) \left (4 x^{3}+18 x^{2}-27 c_{1} +27 x \right )}+5832} \\ y \left (x \right ) &= \frac {\left (\frac {\left (-i \sqrt {3}-1\right ) \left (-36 x^{2}-54 x +108 c_{1} -8 x^{3}+27+6 \sqrt {-6 \left (1+2 c_{1} \right ) \left (4 x^{3}+18 x^{2}-27 c_{1} +27 x \right )}\right )^{\frac {2}{3}}}{4}+\left (2 x +\frac {3}{2}\right ) \left (-36 x^{2}-54 x +108 c_{1} -8 x^{3}+27+6 \sqrt {-6 \left (1+2 c_{1} \right ) \left (4 x^{3}+18 x^{2}-27 c_{1} +27 x \right )}\right )^{\frac {1}{3}}+\left (i \sqrt {3}-1\right ) \left (x +\frac {3}{2}\right )^{2}\right ) {\left (\frac {\left (-36 x^{2}-54 x +108 c_{1} -8 x^{3}+27+6 \sqrt {-6 \left (1+2 c_{1} \right ) \left (4 x^{3}+18 x^{2}-27 c_{1} +27 x \right )}\right )^{\frac {2}{3}} \left (i-\sqrt {3}\right )}{4}-i \left (-x +\frac {3}{2}\right ) \left (-36 x^{2}-54 x +108 c_{1} -8 x^{3}+27+6 \sqrt {-6 \left (1+2 c_{1} \right ) \left (4 x^{3}+18 x^{2}-27 c_{1} +27 x \right )}\right )^{\frac {1}{3}}+\left (x +\frac {3}{2}\right )^{2} \left (\sqrt {3}+i\right )\right )}^{2}}{216 x^{3}+972 x^{2}+1458 x -2916 c_{1} -162 \sqrt {-6 \left (1+2 c_{1} \right ) \left (4 x^{3}+18 x^{2}-27 c_{1} +27 x \right )}-729} \\ y \left (x \right ) &= \frac {\left (\frac {\left (i \sqrt {3}-1\right ) \left (-36 x^{2}-54 x +108 c_{1} -8 x^{3}+27+6 \sqrt {-6 \left (1+2 c_{1} \right ) \left (4 x^{3}+18 x^{2}-27 c_{1} +27 x \right )}\right )^{\frac {2}{3}}}{4}-\left (-2 x -\frac {3}{2}\right ) \left (-36 x^{2}-54 x +108 c_{1} -8 x^{3}+27+6 \sqrt {-6 \left (1+2 c_{1} \right ) \left (4 x^{3}+18 x^{2}-27 c_{1} +27 x \right )}\right )^{\frac {1}{3}}+\left (-i \sqrt {3}-1\right ) \left (x +\frac {3}{2}\right )^{2}\right ) {\left (\frac {\left (-36 x^{2}-54 x +108 c_{1} -8 x^{3}+27+6 \sqrt {-6 \left (1+2 c_{1} \right ) \left (4 x^{3}+18 x^{2}-27 c_{1} +27 x \right )}\right )^{\frac {2}{3}} \left (\sqrt {3}+i\right )}{4}+i \left (x -\frac {3}{2}\right ) \left (-36 x^{2}-54 x +108 c_{1} -8 x^{3}+27+6 \sqrt {-6 \left (1+2 c_{1} \right ) \left (4 x^{3}+18 x^{2}-27 c_{1} +27 x \right )}\right )^{\frac {1}{3}}+\left (x +\frac {3}{2}\right )^{2} \left (i-\sqrt {3}\right )\right )}^{2}}{216 x^{3}+972 x^{2}+1458 x -2916 c_{1} -162 \sqrt {-6 \left (1+2 c_{1} \right ) \left (4 x^{3}+18 x^{2}-27 c_{1} +27 x \right )}-729} \\ \end{align*}
✓ Solution by Mathematica
Time used: 84.456 (sec). Leaf size: 1516
DSolve[-y[x] + x*y'[x]^2 + y'[x]^3==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*}