Internal problem ID [6046]
Book: Elementary differential equations. By Earl D. Rainville, Phillip E. Bedient. Macmilliam
Publishing Co. NY. 6th edition. 1981.
Section: CHAPTER 16. Nonlinear equations. Section 99. Clairaut’s equation. EXERCISES Page
320
Problem number: 7.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_1st_order, _with_linear_symmetries], _Clairaut]
Solve \begin {gather*} \boxed {y-x y^{\prime }-k \left (y^{\prime }\right )^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.17 (sec). Leaf size: 24
dsolve(y(x)=diff(y(x),x)*x+k*diff(y(x),x)^2,y(x), singsol=all)
\begin{align*} y \relax (x ) = -\frac {x^{2}}{4 k} \\ y \relax (x ) = c_{1}^{2} k +c_{1} x \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.011 (sec). Leaf size: 28
DSolve[y[x]==y'[x]*x+k*(y'[x])^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_1 (x+c_1 k) \\ y(x)\to -\frac {x^2}{4 k} \\ \end{align*}