9.8.54 problem 57

Internal problem ID [3057]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 12, page 46
Problem number : 57
Date solved : Tuesday, September 30, 2025 at 06:26:39 AM
CAS classification : [_separable]

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

With initial conditions

\begin{align*} y \left (0\right )&=1 \\ \end{align*}

✓ Maple. Time used: 0.078 (sec). Leaf size: 17

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

\[ y = \frac {1}{\sqrt {2-\sqrt {x^{2}+1}}} \]

✓ Mathematica. Time used: 3.84 (sec). Leaf size: 32

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

\begin{align*} y(x)&\to \frac {\sqrt {-\sqrt {x^2+1}-2}}{\sqrt {x^2-3}} \end{align*}

✓ Sympy. Time used: 8.217 (sec). Leaf size: 22

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

\[ y{\left (x \right )} = \sqrt {\frac {- \sqrt {x^{2} + 1} - 2}{x^{2} - 3}} \]