68.11.40 problem 52

Internal problem ID [17577]
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 : 52
Date solved : Thursday, October 02, 2025 at 02:25:38 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+7 y^{\prime }+10 y&=t \,{\mathrm e}^{-t} \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=-{\frac {5}{16}} \\ y^{\prime }\left (0\right )&={\frac {9}{16}} \\ \end{align*}

✓ Maple. Time used: 0.019 (sec). Leaf size: 15

ode:=diff(diff(y(t),t),t)+7*diff(y(t),t)+10*y(t) = t*exp(-t); 
ic:=[y(0) = -5/16, D(y)(0) = 9/16]; 
dsolve([ode,op(ic)],y(t), singsol=all);
 

\[ y = \frac {\left (4 t -5\right ) {\mathrm e}^{-t}}{16} \]

✓ Mathematica. Time used: 0.014 (sec). Leaf size: 19

ode=D[y[t],{t,2}]+7*D[y[t],t]+10*y[t]==t*Exp[-t]; 
ic={y[0]==-15/48,Derivative[1][y][0] ==9/16}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \frac {1}{16} e^{-t} (4 t-5) \end{align*}

✓ Sympy. Time used: 0.186 (sec). Leaf size: 12

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-t*exp(-t) + 10*y(t) + 7*Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): -5/16, Subs(Derivative(y(t), t), t, 0): 9/16} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (\frac {t}{4} - \frac {5}{16}\right ) e^{- t} \]