87.1.12 problem 13

Internal problem ID [23224]
Book : Ordinary differential equations with modern applications. Ladas, G. E. and Finizio, N. Wadsworth Publishing. California. 1978. ISBN 0-534-00552-7. QA372.F56
Section : Chapter 1. Elementary methods. First order differential equations. Exercise at page 9
Problem number : 13
Date solved : Thursday, October 02, 2025 at 09:24:34 PM
CAS classification : [[_2nd_order, _quadrature]]

\begin{align*} m s^{\prime \prime }&=\frac {g \,t^{2}}{2} \end{align*}

✓ Maple. Time used: 0.001 (sec). Leaf size: 18

ode:=m*diff(diff(s(t),t),t) = 1/2*g*t^2; 
dsolve(ode,s(t), singsol=all);
 

\[ s = \frac {g \,t^{4}}{24 m}+c_1 t +c_2 \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 23

ode=m*D[s[t],{t,2}]==1/2*g*t^2; 
ic={}; 
DSolve[{ode,ic},s[t],t,IncludeSingularSolutions->True]
 

\begin{align*} s(t)&\to \frac {g t^4}{24 m}+c_2 t+c_1 \end{align*}

✓ Sympy. Time used: 0.056 (sec). Leaf size: 15

from sympy import * 
t = symbols("t") 
m = symbols("m") 
g = symbols("g") 
s = Function("s") 
ode = Eq(-g*t**2/2 + m*Derivative(s(t), (t, 2)),0) 
ics = {} 
dsolve(ode,func=s(t),ics=ics)
 

\[ s{\left (t \right )} = C_{1} + C_{2} t + \frac {g t^{4}}{24 m} \]