68.11.45 problem 57

Internal problem ID [17582]
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 : 57
Date solved : Thursday, October 02, 2025 at 02:25:43 PM
CAS classification : [[_2nd_order, _missing_y]]

\begin{align*} y^{\prime \prime }+4 y^{\prime }&=-24 t -6-4 t \,{\mathrm e}^{-4 t}+{\mathrm e}^{-4 t} \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=0 \\ y^{\prime }\left (0\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.031 (sec). Leaf size: 16

ode:=diff(diff(y(t),t),t)+4*diff(y(t),t) = -24*t-6-4*t*exp(-4*t)+exp(-4*t); 
ic:=[y(0) = 0, D(y)(0) = 0]; 
dsolve([ode,op(ic)],y(t), singsol=all);
 

\[ y = t^{2} \left (-3+\frac {{\mathrm e}^{-4 t}}{2}\right ) \]

✓ Mathematica. Time used: 2.539 (sec). Leaf size: 88

ode=D[y[t],{t,2}]+4*D[y[t],t]==-24*t-6-4*t*Exp[-4*t]+Exp[-4*t]; 
ic={y[0]==0,Derivative[1][y][0] ==0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \frac {1}{2} \left (e^{-4 t}-6\right ) t^2\\ y(t)&\to \int _1^t-e^{-4 K[1]} K[1] \left (2 K[1]+6 e^{4 K[1]}-1\right )dK[1]-\int _1^0-e^{-4 K[1]} K[1] \left (2 K[1]+6 e^{4 K[1]}-1\right )dK[1] \end{align*}

✓ Sympy. Time used: 0.200 (sec). Leaf size: 17

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(24*t + 4*t*exp(-4*t) + 4*Derivative(y(t), t) + Derivative(y(t), (t, 2)) + 6 - exp(-4*t),0) 
ics = {y(0): 0, Subs(Derivative(y(t), t), t, 0): 0} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = - 3 t^{2} + \frac {t^{2} e^{- 4 t}}{2} \]