81.1.44 problem 2-42

Internal problem ID [21489]
Book : The Differential Equations Problem Solver. VOL. I. M. Fogiel director. REA, NY. 1978. ISBN 78-63609
Section : Chapter 2. Separable differential equations
Problem number : 2-42
Date solved : Thursday, October 02, 2025 at 07:41:50 PM
CAS classification : [_separable]

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

With initial conditions

\begin{align*} y \left (1\right )&=\frac {\pi }{2} \\ \end{align*}

✓ Maple. Time used: 0.189 (sec). Leaf size: 35

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

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

✓ Mathematica. Time used: 19.689 (sec). Leaf size: 21

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

\begin{align*} y(x)&\to \arcsin \left (\frac {\sqrt {2}}{\sqrt {x^2+1}}\right ) \end{align*}

✓ Sympy. Time used: 0.294 (sec). Leaf size: 36

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

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