78.4.3 problem 3.c

Internal problem ID [21031]
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.c
Date solved : Thursday, October 02, 2025 at 07:01:36 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-y&={\mathrm e}^{2 t} 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: 0.115 (sec). Leaf size: 24

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

\[ y = {\mathrm e}^{t}-\frac {5 \,{\mathrm e}^{-t}}{9}+\frac {\left (3 t -4\right ) {\mathrm e}^{2 t}}{9} \]

✓ Mathematica. Time used: 0.012 (sec). Leaf size: 32

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

\begin{align*} y(t)&\to \frac {1}{9} e^{2 t} (3 t-4)-\frac {5 e^{-t}}{9}+e^t \end{align*}

✓ Sympy. Time used: 0.082 (sec). Leaf size: 24

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-t*exp(2*t) - y(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)
 

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