26.3.12 problem Exact Differential equations. Exercise 9.16, page 79

Internal problem ID [6945]
Book : Ordinary Differential Equations, By Tenenbaum and Pollard. Dover, NY 1963
Section : Chapter 2. Special types of differential equations of the first kind. Lesson 9
Problem number : Exact Differential equations. Exercise 9.16, page 79
Date solved : Tuesday, September 30, 2025 at 04:07:07 PM
CAS classification : [_separable]

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

With initial conditions

\begin{align*} y \left (\frac {\pi }{4}\right )&=\frac {\pi }{4} \\ \end{align*}

✓ Maple. Time used: 0.290 (sec). Leaf size: 9

ode:=sin(x)*cos(y(x))+cos(x)*sin(y(x))*diff(y(x),x) = 0; 
ic:=[y(1/4*Pi) = 1/4*Pi]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = \arccos \left (\frac {\sec \left (x \right )}{2}\right ) \]

✓ Mathematica. Time used: 0.093 (sec). Leaf size: 93

ode=Sin[x]*Cos[y[x]]+Cos[x]*Sin[y[x]]*D[y[x],x]==0; 
ic=y[Pi/4]==Pi/4; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\int _{\frac {\pi }{4}}^x(-\sin (K[1]-y(x))-\sin (K[1]+y(x)))dK[1]+\int _{\frac {\pi }{4}}^{y(x)}\left (\sin (x-K[2])-\sin (x+K[2])-\int _{\frac {\pi }{4}}^x(\cos (K[1]-K[2])-\cos (K[1]+K[2]))dK[1]\right )dK[2]=0,y(x)\right ] \]

✓ Sympy. Time used: 0.368 (sec). Leaf size: 10

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(sin(x)*cos(y(x)) + sin(y(x))*cos(x)*Derivative(y(x), x),0) 
ics = {y(pi/4): pi/4} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \operatorname {acos}{\left (\frac {1}{2 \cos {\left (x \right )}} \right )} \]