76.5.27 problem 28

Internal problem ID [17368]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 2. First order differential equations. Section 2.7 (Substitution Methods). Problems at page 108
Problem number : 28
Date solved : Thursday, March 13, 2025 at 10:04:57 AM
CAS classification : [_Bernoulli]

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

✓ Maple. Time used: 0.057 (sec). Leaf size: 70

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

\begin{align*} y &= -\frac {\sqrt {2}\, \sqrt {\left (x^{2}-1\right ) \left (2 c_{1} -\cos \left (2 x \right )\right )}}{2 x^{2}-2} \\ y &= \frac {\sqrt {2}\, \sqrt {\left (x^{2}-1\right ) \left (2 c_{1} -\cos \left (2 x \right )\right )}}{2 x^{2}-2} \\ \end{align*}

✓ Mathematica. Time used: 0.46 (sec). Leaf size: 56

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

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

✓ Sympy. Time used: 1.353 (sec). Leaf size: 49

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

\[ \left [ y{\left (x \right )} = - \frac {\sqrt {2} \sqrt {\frac {C_{1} - \cos {\left (2 x \right )}}{x^{2} - 1}}}{2}, \ y{\left (x \right )} = \frac {\sqrt {2} \sqrt {\frac {C_{1} - \cos {\left (2 x \right )}}{x^{2} - 1}}}{2}\right ] \]