86.5.13 problem 13

Internal problem ID [23127]
Book : An introduction to Differential Equations. By Howard Frederick Cleaves. 1969. Oliver and Boyd publisher. ISBN 0050015044
Section : Chapter 5. Linear equations of the second order with constant coefficients. Exercise 5a at page 74
Problem number : 13
Date solved : Thursday, October 02, 2025 at 09:23:08 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} 38 x^{\prime \prime }+10 x^{\prime }-3 x&=0 \end{align*}

With initial conditions

\begin{align*} x \left (0\right )&=5 \\ x^{\prime }\left (0\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.050 (sec). Leaf size: 37

ode:=38*diff(diff(x(t),t),t)+10*diff(x(t),t)-3*x(t) = 0; 
ic:=[x(0) = 5, D(x)(0) = 0]; 
dsolve([ode,op(ic)],x(t), singsol=all);
 

\[ x = \frac {5 \,{\mathrm e}^{-\frac {\left (5+\sqrt {139}\right ) t}{38}} \left (\left (139+5 \sqrt {139}\right ) {\mathrm e}^{\frac {t \sqrt {139}}{19}}-5 \sqrt {139}+139\right )}{278} \]

✓ Mathematica. Time used: 0.015 (sec). Leaf size: 54

ode=38*D[x[t],{t,2}]+10*D[x[t],t]-3*x[t]==0; 
ic={x[0]==5,Derivative[1][x][0] ==0}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to \frac {5}{278} e^{-\frac {1}{38} \left (5+\sqrt {139}\right ) t} \left (\left (139+5 \sqrt {139}\right ) e^{\frac {\sqrt {139} t}{19}}+139-5 \sqrt {139}\right ) \end{align*}

✓ Sympy. Time used: 0.141 (sec). Leaf size: 49

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(-3*x(t) + 10*Derivative(x(t), t) + 38*Derivative(x(t), (t, 2)),0) 
ics = {x(0): 5, Subs(Derivative(x(t), t), t, 0): 0} 
dsolve(ode,func=x(t),ics=ics)
 

\[ x{\left (t \right )} = \left (\frac {25 \sqrt {139}}{278} + \frac {5}{2}\right ) e^{\frac {t \left (-5 + \sqrt {139}\right )}{38}} + \left (\frac {5}{2} - \frac {25 \sqrt {139}}{278}\right ) e^{- \frac {t \left (5 + \sqrt {139}\right )}{38}} \]