64.6.16 problem 16

Internal problem ID [13295]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 2, Miscellaneous Review. Exercises page 60
Problem number : 16
Date solved : Monday, March 31, 2025 at 07:47:12 AM
CAS classification : [_separable]

\begin{align*} 2 y^{2}+8+\left (-x^{2}+1\right ) y y^{\prime }&=0 \end{align*}

With initial conditions

\begin{align*} y \left (3\right )&=0 \end{align*}

✓ Maple. Time used: 0.064 (sec). Leaf size: 45

ode:=2*y(x)^2+8+(-x^2+1)*y(x)*diff(y(x),x) = 0; 
ic:=y(3) = 0; 
dsolve([ode,ic],y(x), singsol=all);
 

\begin{align*} y &= -\frac {2 \sqrt {3 x^{2}-10 x +3}}{x +1} \\ y &= \frac {2 \sqrt {3 x^{2}-10 x +3}}{x +1} \\ \end{align*}

✓ Mathematica. Time used: 0.513 (sec). Leaf size: 67

ode=2*(y[x]^2+4)+(1-x^2)*y[x]*D[y[x],x]==0; 
ic={y[3]==0}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to -2 \sqrt {\exp \left (2 \int _3^x\frac {2}{K[1]^2-1}dK[1]\right )-1} \\ y(x)\to 2 \sqrt {\exp \left (2 \int _3^x\frac {2}{K[1]^2-1}dK[1]\right )-1} \\ \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((1 - x**2)*y(x)*Derivative(y(x), x) + 2*y(x)**2 + 8,0) 
ics = {y(3): 0} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : Initial conditions produced too many solutions for constants