Internal problem ID [8103]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 523.
ODE order: 1.
ODE degree: 3.
CAS Maple gives this as type [_quadrature]
Solve \begin {gather*} \boxed {\left (y^{\prime }\right )^{3}-a x y^{\prime }+x^{3}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.2 (sec). Leaf size: 300
dsolve(diff(y(x),x)^3-a*x*diff(y(x),x)+x^3=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = \int \frac {i \left (\sqrt {3}\, \left (-108 x^{3}+12 \sqrt {-3 x^{3} \left (4 a^{3}-27 x^{3}\right )}\right )^{\frac {2}{3}}-12 \sqrt {3}\, a x +i \left (-108 x^{3}+12 \sqrt {-3 x^{3} \left (4 a^{3}-27 x^{3}\right )}\right )^{\frac {2}{3}}+12 i a x \right )}{12 \left (-108 x^{3}+12 \sqrt {-3 x^{3} \left (4 a^{3}-27 x^{3}\right )}\right )^{\frac {1}{3}}}d x +c_{1} \\ y \relax (x ) = \int \frac {i \left (i \left (-108 x^{3}+12 \sqrt {-3 x^{3} \left (4 a^{3}-27 x^{3}\right )}\right )^{\frac {2}{3}}+12 i a x -\sqrt {3}\, \left (-108 x^{3}+12 \sqrt {-3 x^{3} \left (4 a^{3}-27 x^{3}\right )}\right )^{\frac {2}{3}}+12 \sqrt {3}\, a x \right )}{12 \left (-108 x^{3}+12 \sqrt {-3 x^{3} \left (4 a^{3}-27 x^{3}\right )}\right )^{\frac {1}{3}}}d x +c_{1} \\ y \relax (x ) = \int \frac {\left (-108 x^{3}+12 \sqrt {-3 x^{3} \left (4 a^{3}-27 x^{3}\right )}\right )^{\frac {2}{3}}+12 a x}{6 \left (-108 x^{3}+12 \sqrt {-3 x^{3} \left (4 a^{3}-27 x^{3}\right )}\right )^{\frac {1}{3}}}d x +c_{1} \\ \end{align*}
✓ Solution by Mathematica
Time used: 158.432 (sec). Leaf size: 309
DSolve[x^3 - a*x*y'[x] + y'[x]^3==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \int _1^x\frac {2 \sqrt [3]{3} a K[1]+\sqrt [3]{2} \left (\sqrt {81 K[1]^6-12 a^3 K[1]^3}-9 K[1]^3\right )^{2/3}}{6^{2/3} \sqrt [3]{\sqrt {81 K[1]^6-12 a^3 K[1]^3}-9 K[1]^3}}dK[1]+c_1 \\ y(x)\to \int _1^x\frac {\sqrt [3]{-1} \left (\sqrt [3]{-2} \left (\sqrt {81 K[2]^6-12 a^3 K[2]^3}-9 K[2]^3\right )^{2/3}-2 \sqrt [3]{3} a K[2]\right )}{6^{2/3} \sqrt [3]{\sqrt {81 K[2]^6-12 a^3 K[2]^3}-9 K[2]^3}}dK[2]+c_1 \\ y(x)\to \int _1^x\frac {\sqrt [3]{-1} \left (2 \sqrt [3]{-3} a K[3]-\sqrt [3]{2} \left (\sqrt {81 K[3]^6-12 a^3 K[3]^3}-9 K[3]^3\right )^{2/3}\right )}{6^{2/3} \sqrt [3]{\sqrt {81 K[3]^6-12 a^3 K[3]^3}-9 K[3]^3}}dK[3]+c_1 \\ \end{align*}