22.1.39 problem 39

Internal problem ID [4345]
Book : Differential equations for engineers by Wei-Chau XIE, Cambridge Press 2010
Section : Chapter 2. First-Order and Simple Higher-Order Differential Equations. Page 78
Problem number : 39
Date solved : Tuesday, September 30, 2025 at 07:20:49 AM
CAS classification : [`y=_G(x,y')`]

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

✓ Maple. Time used: 0.075 (sec). Leaf size: 21

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

\[ y = -\arcsin \left (\left (x^{2}+1\right ) \left (c_1 +\ln \left (x^{2}+1\right )\right )\right ) \]

✓ Mathematica. Time used: 6.285 (sec). Leaf size: 25

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

\begin{align*} y(x)&\to -\arcsin \left (\left (x^2+1\right ) \left (\log \left (x^2+1\right )+8 c_1\right )\right ) \end{align*}

✓ Sympy. Time used: 1.594 (sec). Leaf size: 37

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

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