47.1.4 problem 4

Internal problem ID [9724]
Book : Elementary differential equations. By Earl D. Rainville, Phillip E. Bedient. Macmilliam Publishing Co. NY. 6th edition. 1981.
Section : CHAPTER 16. Nonlinear equations. Section 94. Factoring the left member. EXERCISES Page 309
Problem number : 4
Date solved : Tuesday, September 30, 2025 at 06:32:21 PM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 19

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

\begin{align*} y &= c_1 x \\ y &= \frac {-x +c_1}{x} \\ \end{align*}

✓ Mathematica. Time used: 0.026 (sec). Leaf size: 31

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

\begin{align*} y(x)&\to c_1 x\\ y(x)&\to -1+\frac {c_1}{x}\\ y(x)&\to -1\\ y(x)&\to 0 \end{align*}

✓ Sympy. Time used: 0.116 (sec). Leaf size: 12

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

\[ \left [ y{\left (x \right )} = C_{1} x, \ y{\left (x \right )} = \frac {C_{1} - x}{x}\right ] \]