15.1.20 problem 20

Internal problem ID [2860]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 5, page 21
Problem number : 20
Date solved : Sunday, March 30, 2025 at 12:34:49 AM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 31

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

\begin{align*} y &= \sqrt {{\mathrm e}^{-2 \arctan \left (x \right )} c_1 +1} \\ y &= -\sqrt {{\mathrm e}^{-2 \arctan \left (x \right )} c_1 +1} \\ \end{align*}

✓ Mathematica. Time used: 0.843 (sec). Leaf size: 55

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

\begin{align*} y(x)\to -\sqrt {1+e^{-2 \arctan (x)+2 c_1}} \\ y(x)\to \sqrt {1+e^{-2 \arctan (x)+2 c_1}} \\ y(x)\to -1 \\ y(x)\to 1 \\ \end{align*}

✓ Sympy. Time used: 0.635 (sec). Leaf size: 32

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

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