53.1.14 problem 14

Internal problem ID [8448]
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 : 14
Date solved : Wednesday, March 05, 2025 at 05:48:08 AM
CAS classification : [_quadrature]

\begin{align*} x y {y^{\prime }}^{2}+\left (x y^{2}-1\right ) y^{\prime }-y&=0 \end{align*}

✓ Maple. Time used: 0.010 (sec). Leaf size: 34

ode:=x*y(x)*diff(y(x),x)^2+(-1+x*y(x)^2)*diff(y(x),x)-y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \sqrt {2 \ln \left (x \right )+c_{1}} \\ y &= -\sqrt {2 \ln \left (x \right )+c_{1}} \\ y &= c_{1} {\mathrm e}^{-x} \\ \end{align*}

✓ Mathematica. Time used: 0.119 (sec). Leaf size: 57

ode=x*y[x]*(D[y[x],x])^2+(x*y[x]^2-1)*D[y[x],x]-y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to c_1 e^{-x} \\ y(x)\to -\sqrt {2} \sqrt {\log (x)+c_1} \\ y(x)\to \sqrt {2} \sqrt {\log (x)+c_1} \\ y(x)\to 0 \\ \end{align*}

✓ Sympy. Time used: 0.600 (sec). Leaf size: 32

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*y(x)*Derivative(y(x), x)**2 + (x*y(x)**2 - 1)*Derivative(y(x), x) - y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = - \sqrt {C_{1} + 2 \log {\left (x \right )}}, \ y{\left (x \right )} = \sqrt {C_{1} + 2 \log {\left (x \right )}}, \ y{\left (x \right )} = C_{1} e^{- x}\right ] \]