76.15.15 problem 16

Internal problem ID [17585]
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 : 16
Date solved : Monday, March 31, 2025 at 04:18:20 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.007 (sec). Leaf size: 43

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

\[ y = \frac {\left (\left (6 c_1 +t \right ) {\mathrm e}^{4 t}+6 c_2 \,{\mathrm e}^{t}-\frac {\cosh \left (3 t \right ) {\mathrm e}^{t}}{3}-\frac {\sinh \left (3 t \right ) {\mathrm e}^{t}}{3}+\frac {3}{4}\right ) {\mathrm e}^{-2 t}}{6} \]

✓ Mathematica. Time used: 0.043 (sec). Leaf size: 39

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

\[ y(t)\to \frac {1}{72} e^{-2 t} \left (72 c_1 e^t+4 e^{4 t} (3 t-1+18 c_2)+9\right ) \]

✗ Sympy

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

NotImplementedError : The given ODE 2*y(t) + cosh(2*t) + Derivative(y(t), t) - Derivative(y(t), (t, 2)) cannot be solved by the factorable group method