7.1.17 problem 17

Internal problem ID [17]
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 : 17
Date solved : Tuesday, March 04, 2025 at 10:38:33 AM
CAS classification : [[_2nd_order, _quadrature]]

\begin{align*} x^{\prime \prime }&=\frac {1}{\left (t +1\right )^{3}} \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.008 (sec). Leaf size: 15

ode:=diff(diff(x(t),t),t) = 1/(t+1)^3; 
ic:=x(0) = 0, D(x)(0) = 0; 
dsolve([ode,ic],x(t), singsol=all);
 

\[ x = \frac {t^{2}}{2 t +2} \]

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 16

ode=D[x[t],{t,2}]==1/(1+t)^3; 
ic={x[0]==0,Derivative[1][x][0] ==0}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\[ x(t)\to \frac {t^2}{2 t+2} \]

✓ Sympy. Time used: 0.242 (sec). Leaf size: 34

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(Derivative(x(t), (t, 2)) - 1/(t + 1)**3,0) 
ics = {x(0): 0, Subs(Derivative(x(t), t), t, 0): 0} 
dsolve(ode,func=x(t),ics=ics)
 

\[ x{\left (t \right )} = t \left (\frac {1}{2} + \frac {1}{2 \left (t^{2} + 2 t + 1\right )}\right ) - \frac {1}{2} + \frac {1}{2 \left (t^{2} + 2 t + 1\right )} \]