68.12.50 problem 58 (a)

Internal problem ID [17644]
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 : 58 (a)
Date solved : Thursday, October 02, 2025 at 02:26:45 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 4 y^{\prime \prime }+4 y^{\prime }+y&={\mathrm e}^{-\frac {t}{2}} \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.024 (sec). Leaf size: 26

ode:=4*diff(diff(y(t),t),t)+4*diff(y(t),t)+y(t) = exp(-1/2*t); 
ic:=[y(0) = a, D(y)(0) = b]; 
dsolve([ode,op(ic)],y(t), singsol=all);
 

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

✓ Mathematica. Time used: 0.016 (sec). Leaf size: 30

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

\begin{align*} y(t)&\to \frac {1}{8} e^{-t/2} (4 a (t+2)+t (8 b+t)) \end{align*}

✓ Sympy. Time used: 0.167 (sec). Leaf size: 19

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

\[ y{\left (t \right )} = \left (a + t \left (\frac {a}{2} + b + \frac {t}{8}\right )\right ) e^{- \frac {t}{2}} \]