12.2.28 problem 28

Internal problem ID [1564]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 2, First order equations. Linear first order. Section 2.1 Page 41
Problem number : 28
Date solved : Saturday, March 29, 2025 at 10:59:19 PM
CAS classification : [_linear]

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

With initial conditions

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

✓ Maple. Time used: 0.026 (sec). Leaf size: 13

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

\[ y = -\frac {\cos \left (x \right ) \cot \left (x \right )}{2}+\csc \left (x \right ) \]

✓ Mathematica. Time used: 0.039 (sec). Leaf size: 16

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

\[ y(x)\to \csc (x)-\frac {1}{2} \cos (x) \cot (x) \]

✓ Sympy. Time used: 1.000 (sec). Leaf size: 14

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

\[ y{\left (x \right )} = \frac {1 - \frac {\cos ^{2}{\left (x \right )}}{2}}{\sin {\left (x \right )}} \]