90.4.12 problem 13

Internal problem ID [25107]
Book : Ordinary Differential Equations. By William Adkins and Mark G Davidson. Springer. NY. 2010. ISBN 978-1-4614-3617-1
Section : Chapter 1. First order differential equations. Exercises at page 59
Problem number : 13
Date solved : Thursday, October 02, 2025 at 11:50:22 PM
CAS classification : [_separable]

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

With initial conditions

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

✓ Maple. Time used: 0.008 (sec). Leaf size: 13

ode:=diff(y(t),t)+y(t)*cos(t) = cos(t); 
ic:=[y(0) = 0]; 
dsolve([ode,op(ic)],y(t), singsol=all);
 

\[ y = 1-{\mathrm e}^{-\sin \left (t \right )} \]

✓ Mathematica. Time used: 0.03 (sec). Leaf size: 15

ode=D[y[t],{t,1}] +y[t]*Cos[t]== Cos[t]; 
ic={y[0]==0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to 1-e^{-\sin (t)} \end{align*}

✓ Sympy. Time used: 0.159 (sec). Leaf size: 10

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

\[ y{\left (t \right )} = 1 - e^{- \sin {\left (t \right )}} \]