57.7.2 problem 1(b)

Internal problem ID [14402]
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 2.2.3 Complex eigenvalues. Exercises page 94
Problem number : 1(b)
Date solved : Thursday, October 02, 2025 at 09:36:43 AM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} x^{\prime \prime }-4 x^{\prime }+6 x&=0 \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.095 (sec). Leaf size: 27

ode:=diff(diff(x(t),t),t)-4*diff(x(t),t)+6*x(t) = 0; 
ic:=[x(0) = 1, D(x)(0) = 0]; 
dsolve([ode,op(ic)],x(t), singsol=all);
 

\[ x = {\mathrm e}^{2 t} \left (-\sqrt {2}\, \sin \left (\sqrt {2}\, t \right )+\cos \left (\sqrt {2}\, t \right )\right ) \]

✓ Mathematica. Time used: 0.013 (sec). Leaf size: 35

ode=D[x[t],{t,2}]-4*D[x[t],t]+6*x[t]==0; 
ic={x[0]==1,Derivative[1][x][0 ]==0}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to e^{2 t} \left (\cos \left (\sqrt {2} t\right )-\sqrt {2} \sin \left (\sqrt {2} t\right )\right ) \end{align*}

✓ Sympy. Time used: 0.113 (sec). Leaf size: 29

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(6*x(t) - 4*Derivative(x(t), t) + Derivative(x(t), (t, 2)),0) 
ics = {x(0): 1, Subs(Derivative(x(t), t), t, 0): 0} 
dsolve(ode,func=x(t),ics=ics)
 

\[ x{\left (t \right )} = \left (- \sqrt {2} \sin {\left (\sqrt {2} t \right )} + \cos {\left (\sqrt {2} t \right )}\right ) e^{2 t} \]