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 : Sunday, March 30, 2025 at 01:11:04 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*}

✓ Maple. Time used: 0.090 (sec). Leaf size: 42

ode:=diff(y(x),x)*(x*diff(y(x),x)-y(x)+k)+a = 0; 
dsolve(ode,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*}

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

ode=D[y[x],x]*( x*D[y[x],x]-y[x]+k )+a==0; 
ic={}; 
DSolve[{ode,ic},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*}

✗ Sympy

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

Timed Out