90.7.17 problem 17

Internal problem ID [25171]
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 99
Problem number : 17
Date solved : Thursday, October 02, 2025 at 11:56:17 PM
CAS classification : [[_homogeneous, `class C`], _dAlembert]

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

With initial conditions

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

✗ Maple

ode:=diff(y(t),t) = cos(y(t)+t); 
ic:=[y(t__0) = y__0]; 
dsolve([ode,op(ic)],y(t), singsol=all);
 

\[ \text {No solution found} \]

✓ Mathematica. Time used: 0.547 (sec). Leaf size: 49

ode=D[y[t],t]== Cos[t+y[t]]; 
ic={y[t0]==y0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to 2 \arctan \left (t+\tan \left (\frac {\text {t0}+\text {y0}}{2}\right )-\text {t0}\right )-t\\ y(t)&\to 2 \arctan \left (t+\tan \left (\frac {\text {t0}+\text {y0}}{2}\right )-\text {t0}\right )-t \end{align*}

✓ Sympy. Time used: 0.596 (sec). Leaf size: 19

from sympy import * 
t = symbols("t") 
t0 = symbols("t0") 
y0 = symbols("y0") 
y = Function("y") 
ode = Eq(-cos(t + y(t)) + Derivative(y(t), t),0) 
ics = {y(t0): y0} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = - t + 2 \operatorname {atan}{\left (t - t_{0} + \tan {\left (\frac {t_{0}}{2} + \frac {y_{0}}{2} \right )} \right )} \]