58.20.9 problem 9

Internal problem ID [14933]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 9, The Laplace transform. Section 9.3, Exercises page 452
Problem number : 9
Date solved : Thursday, October 02, 2025 at 09:56:41 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+7 y^{\prime }+10 y&=4 t \,{\mathrm e}^{-3 t} \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=0 \\ y^{\prime }\left (0\right )&=-1 \\ \end{align*}

✓ Maple. Time used: 0.092 (sec). Leaf size: 16

ode:=diff(diff(y(t),t),t)+7*diff(y(t),t)+10*y(t) = 4*t*exp(-3*t); 
ic:=[y(0) = 0, D(y)(0) = -1]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = \left ({\mathrm e}^{t}-2 t -1\right ) {\mathrm e}^{-3 t} \]

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

ode=D[y[t],{t,2}]+7*D[y[t],t]+10*y[t]==4*t*Exp[-3*t]; 
ic={y[0]==0,Derivative[1][y][0]==-1}; 
DSolve[{ode,ic},{y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to e^{-3 t} \left (-2 t+e^t-1\right ) \end{align*}

✓ Sympy. Time used: 0.214 (sec). Leaf size: 20

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-4*t*exp(-3*t) + 10*y(t) + 7*Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 0, Subs(Derivative(y(t), t), t, 0): -1} 
dsolve(ode,func=y(t),ics=ics)
 

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