85.18.5 problem 1 (e)

Internal problem ID [22551]
Book : Applied Differential Equations. By Murray R. Spiegel. 3rd edition. 1980. Pearson. ISBN 978-0130400970
Section : Chapter two. First order and simple higher order ordinary differential equations. A Exercises at page 52
Problem number : 1 (e)
Date solved : Thursday, October 02, 2025 at 08:50:25 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x)*G(y),0]`]]

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

With initial conditions

\begin{align*} y \left (0\right )&=\frac {\pi }{2} \\ \end{align*}

✓ Maple. Time used: 0.295 (sec). Leaf size: 19

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

\[ y = \operatorname {RootOf}\left (-\sin \left (\textit {\_Z} \right ) \pi +2 \textit {\_Z} \sin \left (\textit {\_Z} \right )+2 x \right ) \]

✓ Mathematica. Time used: 0.175 (sec). Leaf size: 21

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

\[ \text {Solve}\left [x=\frac {1}{2} \pi \sin (y(x))-y(x) \sin (y(x)),y(x)\right ] \]

✓ Sympy. Time used: 3.193 (sec). Leaf size: 14

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

\[ \frac {x}{\sin {\left (y{\left (x \right )} \right )}} + y{\left (x \right )} - \frac {\pi }{2} = 0 \]