76.15.14 problem 15

Internal problem ID [17584]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 4. Second order linear equations. Section 4.5 (Nonhomogeneous Equations, Method of Undetermined Coefficients). Problems at page 260
Problem number : 15
Date solved : Monday, March 31, 2025 at 04:18:18 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+y^{\prime }+4 y&=2 \sinh \left (t \right ) \end{align*}

✓ Maple. Time used: 0.007 (sec). Leaf size: 41

ode:=diff(diff(y(t),t),t)+diff(y(t),t)+4*y(t) = 2*sinh(t); 
dsolve(ode,y(t), singsol=all);
 

\[ y = {\mathrm e}^{-\frac {t}{2}} \sin \left (\frac {\sqrt {15}\, t}{2}\right ) c_2 +{\mathrm e}^{-\frac {t}{2}} \cos \left (\frac {\sqrt {15}\, t}{2}\right ) c_1 -\frac {{\mathrm e}^{-t}}{4}+\frac {{\mathrm e}^{t}}{6} \]

✓ Mathematica. Time used: 0.33 (sec). Leaf size: 64

ode=D[y[t],{t,2}]+D[y[t],t]+4*y[t]==2*Sinh[t]; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to -\frac {e^{-t}}{4}+\frac {e^t}{6}+c_2 e^{-t/2} \cos \left (\frac {\sqrt {15} t}{2}\right )+c_1 e^{-t/2} \sin \left (\frac {\sqrt {15} t}{2}\right ) \]

✓ Sympy. Time used: 0.218 (sec). Leaf size: 42

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(4*y(t) - 2*sinh(t) + Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (C_{1} \sin {\left (\frac {\sqrt {15} t}{2} \right )} + C_{2} \cos {\left (\frac {\sqrt {15} t}{2} \right )}\right ) e^{- \frac {t}{2}} + \frac {5 \sinh {\left (t \right )}}{12} - \frac {\cosh {\left (t \right )}}{12} \]