37.1.3 problem 10.2.8 part(1)

Internal problem ID [6390]
Book : Basic Training in Mathematics. By R. Shankar. Plenum Press. NY. 1995
Section : Chapter 10, Differential equations. Section 10.2, ODEs with constant Coefficients. page 307
Problem number : 10.2.8 part(1)
Date solved : Wednesday, March 05, 2025 at 12:37:21 AM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} x^{\prime \prime }+42 x^{\prime }+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: 0.045 (sec). Leaf size: 43

ode:=diff(diff(x(t),t),t)+42*diff(x(t),t)+x(t) = 0; 
ic:=x(0) = 1, D(x)(0) = 0; 
dsolve([ode,ic],x(t), singsol=all);
 

\[ x \left (t \right ) = \frac {\left (220+21 \sqrt {110}\right ) {\mathrm e}^{\left (-21+2 \sqrt {110}\right ) t}}{440}+\frac {\left (220-21 \sqrt {110}\right ) {\mathrm e}^{\left (-21-2 \sqrt {110}\right ) t}}{440} \]

✓ Mathematica. Time used: 0.032 (sec). Leaf size: 53

ode=D[x[t],{t,2}]+42*D[x[t],t]+x[t]==0; 
ic={x[0]==1,Derivative[1][x][0 ]==0}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\[ x(t)\to \frac {e^{-\left (\left (21+2 \sqrt {110}\right ) t\right )} \left (\left (881+84 \sqrt {110}\right ) e^{4 \sqrt {110} t}-1\right )}{880+84 \sqrt {110}} \]

✓ Sympy. Time used: 0.230 (sec). Leaf size: 49

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(x(t) + 42*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)
 

\[ x{\left (t \right )} = \left (\frac {1}{2} + \frac {21 \sqrt {110}}{440}\right ) e^{t \left (-21 + 2 \sqrt {110}\right )} + \left (\frac {1}{2} - \frac {21 \sqrt {110}}{440}\right ) e^{- t \left (2 \sqrt {110} + 21\right )} \]