68.10.20 problem 20

Internal problem ID [17512]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 4. Higher Order Equations. Exercises 4.2, page 147
Problem number : 20
Date solved : Thursday, October 02, 2025 at 02:24:56 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} 2 y^{\prime \prime }-7 y^{\prime }-4 y&=0 \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.050 (sec). Leaf size: 17

ode:=2*diff(diff(y(t),t),t)-7*diff(y(t),t)-4*y(t) = 0; 
ic:=[y(0) = 0, D(y)(0) = 1]; 
dsolve([ode,op(ic)],y(t), singsol=all);
 

\[ y = \frac {2 \,{\mathrm e}^{4 t}}{9}-\frac {2 \,{\mathrm e}^{-\frac {t}{2}}}{9} \]

✓ Mathematica. Time used: 0.01 (sec). Leaf size: 25

ode=2*D[y[t],{t,2}]-7*D[y[t],t]-4*y[t]==0; 
ic={y[0]==0,Derivative[1][y][0] ==1}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \frac {2}{9} e^{-t/2} \left (e^{9 t/2}-1\right ) \end{align*}

✓ Sympy. Time used: 0.106 (sec). Leaf size: 19

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-4*y(t) - 7*Derivative(y(t), t) + 2*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)
 

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