78.4.5 problem 3.e

Internal problem ID [21033]
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.e
Date solved : Thursday, October 02, 2025 at 07:01:37 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.142 (sec). Leaf size: 37

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

\[ y = -\frac {\left (13 \sqrt {17}\, \sinh \left (\frac {t \sqrt {17}}{2}\right )-85 \cosh \left (\frac {t \sqrt {17}}{2}\right )\right ) {\mathrm e}^{\frac {3 t}{2}}}{68}-\frac {{\mathrm e}^{t}}{4} \]

✓ Mathematica. Time used: 0.089 (sec). Leaf size: 76

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

\begin{align*} y(t)&\to \frac {1}{136} e^{-\frac {1}{2} \left (1+\sqrt {17}\right ) t} \left (\left (85+13 \sqrt {17}\right ) e^{2 t}+\left (85-13 \sqrt {17}\right ) e^{\left (2+\sqrt {17}\right ) t}-34 e^{\frac {1}{2} \left (3+\sqrt {17}\right ) t}\right ) \end{align*}

✓ Sympy. Time used: 0.143 (sec). Leaf size: 54

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

\[ y{\left (t \right )} = - \frac {e^{t}}{4} + \left (\frac {13 \sqrt {17}}{136} + \frac {5}{8}\right ) e^{\frac {t \left (3 - \sqrt {17}\right )}{2}} + \left (\frac {5}{8} - \frac {13 \sqrt {17}}{136}\right ) e^{\frac {t \left (3 + \sqrt {17}\right )}{2}} \]