68.12.19 problem 19

Internal problem ID [17613]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 4. Higher Order Equations. Exercises 4.4, page 163
Problem number : 19
Date solved : Thursday, October 02, 2025 at 02:26:16 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-y&=2 \sinh \left (t \right ) \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 28

ode:=diff(diff(y(t),t),t)-y(t) = 2*sinh(t); 
dsolve(ode,y(t), singsol=all);
 

\[ y = \frac {{\mathrm e}^{-t} \left (4 c_2 +2 t \right )}{4}+\frac {{\mathrm e}^{t} \left (t +2 c_1 -\frac {1}{2}\right )}{2} \]

✓ Mathematica. Time used: 0.027 (sec). Leaf size: 38

ode=D[y[t],{t,2}]-y[t]==2*Sinh[t]; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \frac {1}{4} e^{-t} \left (2 t+e^{2 t} (2 t-1+4 c_1)+1+4 c_2\right ) \end{align*}

✓ Sympy. Time used: 0.047 (sec). Leaf size: 17

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-y(t) - 2*sinh(t) + Derivative(y(t), (t, 2)),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = C_{1} e^{- t} + C_{2} e^{t} + t \cosh {\left (t \right )} \]