53.3.11 problem 13

Internal problem ID [8473]
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 : 13
Date solved : Monday, January 27, 2025 at 04:06:36 PM
CAS classification : [[_1st_order, _with_linear_symmetries], _rational, _Clairaut]

\begin{align*} y^{\prime } \left (x y^{\prime }-y+k \right )+a&=0 \end{align*}

✓ Solution by Maple

Time used: 0.059 (sec). Leaf size: 42

dsolve(diff(y(x),x)*( x*diff(y(x),x)-y(x)+k )+a=0,y(x), singsol=all)
 

\begin{align*} y &= k -2 \sqrt {a x} \\ y &= k +2 \sqrt {a x} \\ y &= \frac {c_{1}^{2} x +c_{1} k +a}{c_{1}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.017 (sec). Leaf size: 58

DSolve[D[y[x],x]*( x*D[y[x],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*}