90.26.9 problem 33

Internal problem ID [25410]
Book : Ordinary Differential Equations. By William Adkins and Mark G Davidson. Springer. NY. 2010. ISBN 978-1-4614-3617-1
Section : Chapter 6. Discontinuous Functions and the Laplace Transform. Exercises at page 379
Problem number : 33
Date solved : Friday, October 03, 2025 at 12:01:15 AM
CAS classification : [_quadrature]

\begin{align*} y^{\prime }&=\left \{\begin {array}{cc} 0 & t =0 \\ \sin \left (\frac {1}{t}\right ) & \operatorname {otherwise} \end {array}\right . \end{align*}

Using Laplace method

✗ Maple

ode:=diff(y(t),t) = piecewise(t = 0,0,sin(1/t)); 
dsolve(ode,y(t),method='laplace');
 

\[ \text {No solution found} \]

✓ Mathematica. Time used: 0.187 (sec). Leaf size: 45

ode=D[y[t],{t,1}]==Piecewise[{ {0,t==0}, {Sin[1/t],True}}]; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \left ( \begin {array}{cc} \{ & \begin {array}{cc} t \sin \left (\frac {1}{t}\right )-\operatorname {CosIntegral}\left (\frac {1}{t}\right ) & t\leq 0 \\ -\operatorname {CosIntegral}\left (\frac {1}{t}\right )+t \sin \left (\frac {1}{t}\right )-i \pi & \text {True} \\ \end {array} \\ \end {array} \right )+c_1 \end{align*}

✓ Sympy. Time used: 0.548 (sec). Leaf size: 29

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

\[ y{\left (t \right )} = C_{1} + t \sin {\left (\frac {1}{t} \right )} - \frac {\log {\left (\frac {1}{t^{2}} \right )}}{2} + \log {\left (\frac {1}{t} \right )} - \operatorname {Ci}{\left (\frac {1}{t} \right )} \]