32.2.28 problem 29

Internal problem ID [7745]
Book : Engineering Mathematics. By K. A. Stroud. 5th edition. Industrial press Inc. NY. 2001
Section : Program 24. First order differential equations. Further problems 24. page 1068
Problem number : 29
Date solved : Tuesday, September 30, 2025 at 05:04:19 PM
CAS classification : [_Bernoulli]

\begin{align*} y^{\prime }-y \cot \left (x \right )&=y^{2} \sec \left (x \right )^{2} \end{align*}

With initial conditions

\begin{align*} y \left (\frac {\pi }{4}\right )&=-1 \\ \end{align*}

✓ Maple. Time used: 0.758 (sec). Leaf size: 18

ode:=diff(y(x),x)-y(x)*cot(x) = y(x)^2*sec(x)^2; 
ic:=[y(1/4*Pi) = -1]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = \frac {2 \sin \left (x \right )}{\sqrt {2}-2 \sec \left (x \right )} \]

✓ Mathematica. Time used: 0.293 (sec). Leaf size: 22

ode=D[y[x],x]-y[x]*Cot[x]==y[x]^2*Sec[x]^2; 
ic={y[Pi/4]==-1}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {\sin (2 x)}{\sqrt {2} \cos (x)-2} \end{align*}

✓ Sympy. Time used: 0.208 (sec). Leaf size: 17

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-y(x)**2/cos(x)**2 - y(x)/tan(x) + Derivative(y(x), x),0) 
ics = {y(pi/4): -1} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \frac {\sin {\left (2 x \right )}}{\sqrt {2} \cos {\left (x \right )} - 2} \]