74.8.31 problem 31

Internal problem ID [16096]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 2. First Order Equations. Review exercises, page 80
Problem number : 31
Date solved : Monday, March 31, 2025 at 02:41:49 PM
CAS classification : [[_homogeneous, `class C`], _exact, _dAlembert]

\begin{align*} \cos \left (t -y\right )+\left (1-\cos \left (t -y\right )\right ) y^{\prime }&=0 \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.125 (sec). Leaf size: 19

ode:=cos(t-y(t))+(1-cos(t-y(t)))*diff(y(t),t) = 0; 
ic:=y(Pi) = Pi; 
dsolve([ode,ic],y(t), singsol=all);
 

\[ y = t -\operatorname {RootOf}\left (\textit {\_Z} -t +\pi -\sin \left (\textit {\_Z} \right )\right ) \]

✓ Mathematica. Time used: 0.11 (sec). Leaf size: 59

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

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

✓ Sympy. Time used: 4.652 (sec). Leaf size: 12

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

\[ y{\left (t \right )} + \sin {\left (t - y{\left (t \right )} \right )} - \pi = 0 \]