86.1.9 problem 9

Internal problem ID [23071]
Book : An introduction to Differential Equations. By Howard Frederick Cleaves. 1969. Oliver and Boyd publisher. ISBN 0050015044
Section : Chapter 3. Some standard types of differential equations. Exercise 3b at page 43
Problem number : 9
Date solved : Thursday, October 02, 2025 at 09:18:48 PM
CAS classification : [_separable]

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

With initial conditions

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

✓ Maple. Time used: 0.096 (sec). Leaf size: 35

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

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

✓ Mathematica. Time used: 33.006 (sec). Leaf size: 19

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

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

✓ Sympy. Time used: 0.336 (sec). Leaf size: 36

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

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