74.11.29 problem 41

Internal problem ID [16200]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 4. Higher Order Equations. Exercises 4.3, page 156
Problem number : 41
Date solved : Thursday, March 13, 2025 at 07:56:16 AM
CAS classification : [[_2nd_order, _missing_y]]

\begin{align*} y^{\prime \prime }+4 y^{\prime }&=8 \,{\mathrm e}^{4 t}-4 \,{\mathrm e}^{-4 t} \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 26

ode:=diff(diff(y(t),t),t)+4*diff(y(t),t) = 8*exp(4*t)-4*exp(-4*t); 
dsolve(ode,y(t), singsol=all);
 

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

✓ Mathematica. Time used: 2.185 (sec). Leaf size: 74

ode=D[y[t],{t,2}]+4*D[y[t],t]==8*Exp[4*t]-4*Exp[-4*t]; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)\to \int _1^te^{-4 K[1]} \left (c_1+e^{8 K[1]}-4 K[1]\right )dK[1]+c_2 \\ y(t)\to \frac {1}{4} \left (e^{-4 t} (4 t+1)+e^{4 t}-e^4-\frac {5}{e^4}+4 c_2\right ) \\ \end{align*}

✗ Sympy

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-8*exp(4*t) + 4*Derivative(y(t), t) + Derivative(y(t), (t, 2)) + 4*exp(-4*t),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

NotImplementedError : The given ODE -3*sinh(4*t) - cosh(4*t) + Derivative(y(t), t) + Derivative(y(t), (t, 2))/4 cannot be solved by the factorable group method