47.3.4 problem 6

Internal problem ID [9752]
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 : 6
Date solved : Tuesday, September 30, 2025 at 06:32:49 PM
CAS classification : [[_1st_order, _with_linear_symmetries], _Clairaut]

\begin{align*} {y^{\prime }}^{2}-x y^{\prime }+y&=0 \end{align*}

✓ Maple. Time used: 0.031 (sec). Leaf size: 19

ode:=diff(y(x),x)^2-x*diff(y(x),x)+y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \frac {x^{2}}{4} \\ y &= c_1 \left (-c_1 +x \right ) \\ \end{align*}

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 25

ode=(D[y[x],x])^2-x*D[y[x],x]+y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to c_1 (x-c_1)\\ y(x)&\to \frac {x^2}{4} \end{align*}

✓ Sympy. Time used: 1.162 (sec). Leaf size: 14

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x*Derivative(y(x), x) + y(x) + Derivative(y(x), x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \frac {x^{2}}{4} - \frac {\left (C_{1} + x\right )^{2}}{4} \]