76.3.3 problem 3

Internal problem ID [17303]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 2. First order differential equations. Section 2.4 (Differences between linear and nonlinear equations). Problems at page 79
Problem number : 3
Date solved : Monday, March 31, 2025 at 03:50:09 PM
CAS classification : [_linear]

\begin{align*} y^{\prime }+\tan \left (t \right ) y&=\sin \left (t \right ) \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.032 (sec). Leaf size: 16

ode:=diff(y(t),t)+tan(t)*y(t) = sin(t); 
ic:=y(Pi) = 0; 
dsolve([ode,ic],y(t), singsol=all);
 

\[ y = \left (-\ln \left (\cos \left (t \right )\right )+i \pi \right ) \cos \left (t \right ) \]

✓ Mathematica. Time used: 0.053 (sec). Leaf size: 20

ode=D[y[t],t]+Tan[t]*y[t]==Sin[t]; 
ic={y[Pi]==0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to i \cos (t) (\pi +i \log (\cos (t))) \]

✓ Sympy. Time used: 0.451 (sec). Leaf size: 14

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(y(t)*tan(t) - sin(t) + Derivative(y(t), t),0) 
ics = {y(pi): 0} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (- \log {\left (\cos {\left (t \right )} \right )} + i \pi \right ) \cos {\left (t \right )} \]