Internal problem ID [2718]
Book: Differential equations for engineers by Wei-Chau XIE, Cambridge Press 2010
Section: Chapter 2. First-Order and Simple Higher-Order Differential Equations. Page
78
Problem number: 85.
ODE order: 1.
ODE degree: 3.
CAS Maple gives this as type [[_1st_order, _with_linear_symmetries], _Clairaut]
Solve \begin {gather*} \boxed {2 \left (y^{\prime }\right )^{2} \left (y-y^{\prime } x \right )-1=0} \end {gather*}
✓ Solution by Maple
Time used: 0.25 (sec). Leaf size: 57
dsolve(2*(diff(y(x),x))^2*(y(x)-x*diff(y(x),x))=1,y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {3 x^{\frac {2}{3}}}{2} \\ y \relax (x ) = -\frac {3 x^{\frac {2}{3}}}{4}-\frac {3 i \sqrt {3}\, x^{\frac {2}{3}}}{4} \\ y \relax (x ) = -\frac {3 x^{\frac {2}{3}}}{4}+\frac {3 i \sqrt {3}\, x^{\frac {2}{3}}}{4} \\ y \relax (x ) = c_{1} x +\frac {1}{2 c_{1}^{2}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.014 (sec). Leaf size: 67
DSolve[2*(y'[x])^2*(y[x]-x*y'[x])==1,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_1 x+\frac {1}{2 c_1{}^2} \\ y(x)\to \frac {3 x^{2/3}}{2} \\ y(x)\to -\frac {3}{2} \sqrt [3]{-1} x^{2/3} \\ y(x)\to \frac {3}{2} (-1)^{2/3} x^{2/3} \\ \end{align*}