89.30.5 problem 7

Internal problem ID [24922]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 16. Equations of order one and higher degree. Exercises at page 243
Problem number : 7
Date solved : Thursday, October 02, 2025 at 10:49:45 PM
CAS classification : [[_1st_order, _with_linear_symmetries], _Clairaut]

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

✓ Maple. Time used: 0.015 (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: 0.005 (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: 1.108 (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)
 

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