84.38.8 problem 26.20

Internal problem ID [22362]
Book : Schaums outline series. Differential Equations By Richard Bronson. 1973. McGraw-Hill Inc. ISBN 0-07-008009-7
Section : Chapter 26. Solutions of linear differential equations with constant coefficients by Laplace transform. Supplementary problems
Problem number : 26.20
Date solved : Thursday, October 02, 2025 at 08:37:56 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-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: 0.039 (sec). Leaf size: 20

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

\[ y = \frac {3 \,{\mathrm e}^{-t}}{4}+\frac {{\mathrm e}^{t} \left (1+2 t \right )}{4} \]

✓ Mathematica. Time used: 0.012 (sec). Leaf size: 27

ode=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}{4} e^{-t} \left (e^{2 t} (2 t+1)+3\right ) \end{align*}

✓ Sympy. Time used: 0.064 (sec). Leaf size: 19

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

\[ y{\left (t \right )} = \left (\frac {t}{2} + \frac {1}{4}\right ) e^{t} + \frac {3 e^{- t}}{4} \]