7.1.16 problem 16

Internal problem ID [16]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 1. First order differential equations. Section 1.2. Problems at page 17
Problem number : 16
Date solved : Saturday, March 29, 2025 at 04:25:52 PM
CAS classification : [[_2nd_order, _quadrature]]

\begin{align*} x^{\prime \prime }&=\frac {1}{\sqrt {t +4}} \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.035 (sec). Leaf size: 16

ode:=diff(diff(x(t),t),t) = 1/(t+4)^(1/2); 
ic:=x(0) = 1, D(x)(0) = -1; 
dsolve([ode,ic],x(t), singsol=all);
 

\[ x = \frac {4 \left (t +4\right )^{{3}/{2}}}{3}-5 t -\frac {29}{3} \]

✓ Mathematica. Time used: 0.011 (sec). Leaf size: 23

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

\[ x(t)\to \frac {1}{3} \left (4 (t+4)^{3/2}-15 t-29\right ) \]

✓ Sympy. Time used: 0.508 (sec). Leaf size: 29

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

\[ x{\left (t \right )} = t \left (\frac {4 \sqrt {t + 4}}{3} - 5\right ) + \frac {16 \sqrt {t + 4}}{3} - \frac {29}{3} \]