68.18.37 problem 43

Internal problem ID [17881]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 4. Higher Order Equations. Chapter 4 review exercises, page 219
Problem number : 43
Date solved : Thursday, October 02, 2025 at 02:29:11 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

With initial conditions

\begin{align*} y \left (0\right )&=0 \\ y^{\prime }\left (0\right )&=4 \\ \end{align*}

✓ Maple. Time used: 0.025 (sec). Leaf size: 20

ode:=diff(diff(y(t),t),t)+3*diff(y(t),t)-4*y(t) = exp(t); 
ic:=[y(0) = 0, D(y)(0) = 4]; 
dsolve([ode,op(ic)],y(t), singsol=all);
 

\[ y = -\frac {19 \,{\mathrm e}^{-4 t}}{25}+\frac {\left (5 t +19\right ) {\mathrm e}^{t}}{25} \]

✓ Mathematica. Time used: 0.016 (sec). Leaf size: 27

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

\begin{align*} y(t)&\to \frac {1}{25} e^{-4 t} \left (e^{5 t} (5 t+19)-19\right ) \end{align*}

✓ Sympy. Time used: 0.147 (sec). Leaf size: 20

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

\[ y{\left (t \right )} = \left (\frac {t}{5} + \frac {19}{25}\right ) e^{t} - \frac {19 e^{- 4 t}}{25} \]