63.6.7 problem 3(c)

Internal problem ID [13047]
Book : A First Course in Differential Equations by J. David Logan. Third Edition. Springer-Verlag, NY. 2015.
Section : Chapter 2, Second order linear equations. Section 2.2.2 Real eigenvalues. Exercises page 90
Problem number : 3(c)
Date solved : Monday, March 31, 2025 at 07:32:45 AM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} \frac {x^{\prime \prime }}{2}+x^{\prime }+\frac {x}{2}&=0 \end{align*}

With initial conditions

\begin{align*} x \left (0\right )&=-1\\ x^{\prime }\left (0\right )&=2 \end{align*}

✓ Maple. Time used: 0.043 (sec). Leaf size: 12

ode:=1/2*diff(diff(x(t),t),t)+diff(x(t),t)+1/2*x(t) = 0; 
ic:=x(0) = -1, D(x)(0) = 2; 
dsolve([ode,ic],x(t), singsol=all);
 

\[ x = {\mathrm e}^{-t} \left (-1+t \right ) \]

✓ Mathematica. Time used: 0.014 (sec). Leaf size: 14

ode=1/2*D[x[t],{t,2}]+D[x[t],t]+1/2*x[t]==0; 
ic={x[0]==-1,Derivative[1][x][0 ]==2}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\[ x(t)\to e^{-t} (t-1) \]

✓ Sympy. Time used: 0.146 (sec). Leaf size: 8

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

\[ x{\left (t \right )} = \left (t - 1\right ) e^{- t} \]