82.4.19 problem 28-19

Internal problem ID [21837]
Book : The Differential Equations Problem Solver. VOL. II. M. Fogiel director. REA, NY. 1978. ISBN 78-63609
Section : Chapter 28. Laplace transforms. Page 850
Problem number : 28-19
Date solved : Thursday, October 02, 2025 at 08:02:43 PM
CAS classification : [[_3rd_order, _missing_y]]

\begin{align*} y^{\prime \prime \prime }+y^{\prime }&={\mathrm e}^{t} \end{align*}

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.046 (sec). Leaf size: 18

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

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

✓ Mathematica. Time used: 0.034 (sec). Leaf size: 20

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

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

✓ Sympy. Time used: 0.101 (sec). Leaf size: 19

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

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