Internal problem ID [10208]
Book: An elementary treatise on differential equations by Abraham Cohen. DC heath publishers.
1906
Section: Chapter IV, differential equations of the first order and higher degree than the first. Article
27. Clairaut equation. Page 56
Problem number: Ex 7.
ODE order: 1.
ODE degree: 3.
CAS Maple gives this as type [[_1st_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y-2 y^{\prime } x -y^{2} \left (y^{\prime }\right )^{3}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.078 (sec). Leaf size: 107
dsolve(y(x)=2*diff(y(x),x)*x+y(x)^2*diff(y(x),x)^3,y(x), singsol=all)
\begin{align*} y \relax (x ) = -\frac {2 \,2^{\frac {1}{4}} 3^{\frac {1}{4}} \left (-x^{3}\right )^{\frac {1}{4}}}{3} \\ y \relax (x ) = \frac {2 \,2^{\frac {1}{4}} 3^{\frac {1}{4}} \left (-x^{3}\right )^{\frac {1}{4}}}{3} \\ y \relax (x ) = -\frac {2 i 2^{\frac {1}{4}} 3^{\frac {1}{4}} \left (-x^{3}\right )^{\frac {1}{4}}}{3} \\ y \relax (x ) = \frac {2 i 2^{\frac {1}{4}} 3^{\frac {1}{4}} \left (-x^{3}\right )^{\frac {1}{4}}}{3} \\ y \relax (x ) = 0 \\ y \relax (x ) = \sqrt {c_{1}^{3}+2 x c_{1}} \\ y \relax (x ) = -\sqrt {c_{1}^{3}+2 x c_{1}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.105 (sec). Leaf size: 119
DSolve[y[x]==2*y'[x]*x+y[x]^2*(y'[x])^3,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\sqrt {2 c_1 x+c_1{}^3} \\ y(x)\to \sqrt {2 c_1 x+c_1{}^3} \\ y(x)\to (-1-i) \left (\frac {2}{3}\right )^{3/4} x^{3/4} \\ y(x)\to (1-i) \left (\frac {2}{3}\right )^{3/4} x^{3/4} \\ y(x)\to (-1+i) \left (\frac {2}{3}\right )^{3/4} x^{3/4} \\ y(x)\to (1+i) \left (\frac {2}{3}\right )^{3/4} x^{3/4} \\ \end{align*}