Internal problem ID [6052]
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: 13.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_1st_order, _with_linear_symmetries], _rational, _Clairaut]
Solve \begin {gather*} \boxed {y^{\prime } \left (x y^{\prime }-y+k \right )+a=0} \end {gather*}
✓ Solution by Maple
Time used: 0.175 (sec). Leaf size: 27
dsolve(diff(y(x),x)*( x*diff(y(x),x)-y(x)+k )+a=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = c_{1} x +\frac {c_{1} k +a}{c_{1}} \\ y \relax (x ) = c_{1} \sqrt {x}+k \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.013 (sec). Leaf size: 58
DSolve[y'[x]*( x*y'[x]-y[x]+k )+a==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {a}{c_1}+k+c_1 x \\ y(x)\to \text {Indeterminate} \\ y(x)\to k-2 \sqrt {a} \sqrt {x} \\ y(x)\to 2 \sqrt {a} \sqrt {x}+k \\ \end{align*}