63.3.3 problem 3

Internal problem ID [12959]
Book : A First Course in Differential Equations by J. David Logan. Third Edition. Springer-Verlag, NY. 2015.
Section : Chapter 1, First order differential equations. Section 1.2 Antiderivatives. Exercises page 19
Problem number : 3
Date solved : Wednesday, March 05, 2025 at 08:54:55 PM
CAS classification : [[_2nd_order, _quadrature]]

\begin{align*} x^{\prime \prime }&=-3 \sqrt {t} \end{align*}

With initial conditions

\begin{align*} x \left (1\right )&=4\\ x^{\prime }\left (1\right )&=2 \end{align*}

✓ Maple. Time used: 0.022 (sec). Leaf size: 14

ode:=diff(diff(x(t),t),t) = -3*t^(1/2); 
ic:=x(1) = 4, D(x)(1) = 2; 
dsolve([ode,ic],x(t), singsol=all);
 

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

✓ Mathematica. Time used: 0.007 (sec). Leaf size: 19

ode=D[x[t],{t,2}]==-3*Sqrt[t]; 
ic={x[1]==4,Derivative[1][x][1 ]==2}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\[ x(t)\to -\frac {4}{5} \left (t^{5/2}-5 t-1\right ) \]

✓ Sympy. Time used: 0.300 (sec). Leaf size: 17

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(3*sqrt(t) + Derivative(x(t), (t, 2)),0) 
ics = {x(1): 4, Subs(Derivative(x(t), t), t, 1): 2} 
dsolve(ode,func=x(t),ics=ics)
 

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