53.3.5 problem 7

Internal problem ID [8467]
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 equation. EXERCISES Page 320
Problem number : 7
Date solved : Monday, January 27, 2025 at 04:06:24 PM
CAS classification : [[_1st_order, _with_linear_symmetries], _Clairaut]

\begin{align*} y&=x y^{\prime }+k {y^{\prime }}^{2} \end{align*}

✓ Solution by Maple

Time used: 0.038 (sec). Leaf size: 22

dsolve(y(x)=diff(y(x),x)*x+k*diff(y(x),x)^2,y(x), singsol=all)
 

\begin{align*} y &= -\frac {x^{2}}{4 k} \\ y &= c_{1} \left (c_{1} k +x \right ) \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.009 (sec). Leaf size: 28

DSolve[y[x]==D[y[x],x]*x+k*(D[y[x],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*}