74.12.50 problem 58 (a)

Internal problem ID [16357]
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 : Tuesday, January 28, 2025 at 09:06:48 AM
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*}

✓ Solution by Maple

Time used: 0.029 (sec). Leaf size: 26

dsolve([4*diff(y(t),t$2)+4*diff(y(t),t)+y(t)=exp(-t/2),y(0) = a, D(y)(0) = b],y(t), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.026 (sec). Leaf size: 30

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

\[ y(t)\to \frac {1}{8} e^{-t/2} (4 a (t+2)+t (8 b+t)) \]