78.4.4 problem 3.d

Internal problem ID [21032]
Book : A FIRST COURSE IN DIFFERENTIAL EQUATIONS FOR SCIENTISTS AND ENGINEERS. By Russell Herman. University of North Carolina Wilmington. LibreText. compiled on 06/09/2025
Section : Chapter 5, Laplace transforms. Problems section 5.7
Problem number : 3.d
Date solved : Thursday, October 02, 2025 at 07:01:36 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-3 y^{\prime }-4 y&=t^{2} \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=2 \\ y^{\prime }\left (0\right )&=1 \\ \end{align*}

✓ Maple. Time used: 0.108 (sec). Leaf size: 26

ode:=diff(diff(y(t),t),t)-3*diff(y(t),t)-4*y(t) = t^2; 
ic:=[y(0) = 2, D(y)(0) = 1]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = \frac {3 t}{8}+\frac {97 \,{\mathrm e}^{4 t}}{160}-\frac {t^{2}}{4}+\frac {9 \,{\mathrm e}^{-t}}{5}-\frac {13}{32} \]

✓ Mathematica. Time used: 0.011 (sec). Leaf size: 33

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

\begin{align*} y(t)&\to \frac {1}{160} \left (-40 t^2+60 t+288 e^{-t}+97 e^{4 t}-65\right ) \end{align*}

✓ Sympy. Time used: 0.136 (sec). Leaf size: 31

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

\[ y{\left (t \right )} = - \frac {t^{2}}{4} + \frac {3 t}{8} + \frac {97 e^{4 t}}{160} - \frac {13}{32} + \frac {9 e^{- t}}{5} \]