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 : Sunday, March 30, 2025 at 01:10:51 PM
CAS classification : [[_1st_order, _with_linear_symmetries], _Clairaut]

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

✓ Maple. Time used: 0.035 (sec). Leaf size: 22

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

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

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

ode=y[x]==D[y[x],x]*x+k*(D[y[x],x])^2; 
ic={}; 
DSolve[{ode,ic},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*}

✓ Sympy. Time used: 1.728 (sec). Leaf size: 12

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

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